equilibriumselectionforsymmetriccoordination ... · equilibriumselectionforsymmetriccoordination...

Equilibrium selection for symmetric coordinationgames with an application to the minimum-effort

game∗

Jun Honda†

February 28, 2012

Abstract

We consider the class of symmetric two-player games that have the prop-erty that for any mixed strategy of the opponent, a player’s best responsesare included in the support of this mixed strategy—the total bandwagon orcoordination property (CP). We show that for any number of pure strategiesn, a symmetric two-player game has CP if and only if the game has 2n − 1symmetric Nash equilibria. In view of the importance of 1/2–dominance asan equilibrium selection criterion, we show that if in addition to CP a gameis supermodular, it will always have a 1/2–dominant equilibrium, and the1/2–dominant equilibrium will be either the lowest or the highest strategyprofile. Furthermore we show that if a game with CP has a unique potentialmaximizer, it will be equivalent to a 1/2–dominant equilibrium. As a specificapplication, we consider the minimum-effort game and reexamine the exper-imental equilibrium selection result of Van Huyck, Battalio and Beil (1990)to compare it with the theoretical prediction. Our results allow us to give anew insight into an aspect of the experiment that so far could not be explained.

KEYWORDS: Equilibrium selection; Coordination games; 1/2–Dominance;Supermodularity; Potential games; Minimum-effort game.

1 Introduction

We consider the class of symmetric two-player games which satisfy the coordina-tion property. The coordination property CP means that, given any opponent’smixed strategy, all best responses against the opponent’s strategy for a player are

∗I am very grateful to Josef Hofbauer for many suggestions and discussions and for constructiveand insightful comments. I would also like to thank Boyu Zhang, Christina Pawlowitsch, DaisukeOyama, Karl Schlag, Klaus Ritzberger, Maarten Janssen, Naoki Yoshihara, Wieland Muller, Ya-suhiro Shirata, and audiences at Micro Research Seminar of Vienna Graduate School of Economicsfor valuable comments and discussions.

†Vienna Graduate School of Economics, Maria Theresien Strasse 3/18, 1090 Vienna, Austria.E-mail: [email protected]

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included in the support of the opponent’s mixed strategy. CP is the same as the“total bandwagon property” given by Kandori and Rob (1998), which is used in aneconomic framework to describe technology adoption.1 The class of games with CPincludes pure coordination games, whereas it is a subset of the class of all coordina-tion games.2 For this class of games, we provide the following characterization: forany number of pure strategies n, a symmetric game has CP if and only if the gamehas 2n−1 symmetric Nash equilibria. This implies that the equilibrium selection forgames with CP in the class of all symmetric coordination games is the most difficultbecause the maximum number of symmetric Nash equilibria in generic symmetrictwo-player games is 2n−1 and any generic game with CP has 2n−1 Nash equilibria.

We take into account two aspects for a class of games with CP and providea simple condition under which a specific equilibrium among multiple equilibria isselected by many equilibrium selection methods. As the first aspect, we considerstrategic complementarity (supermodularity), which has been considered to be im-portant in economics (Milgrom and Roberts, 1990; Milgrom and Shannon, 1994).As the second aspect in terms of the equilibrium selection methods, we also con-sider the solution concept of 1/2–dominant equilibrium proposed by Morris, Rob andShin (1995), which is known as a sufficient condition for various equilibrium selec-tion methods including the “evolutionary learning methods” (Kandori, Mailath andRob, 1993; Young, 1993); the “global game method” (Carlsson and van Damme,1993); the “incomplete information method” (Kajii and Morris, 1997); the “per-fect foresight dynamics method” (Matsui and Matsuyama, 1995); and the “spatialdominance method” (Hofbauer, 1999; Deguchi and Hofbauer, 2010). Taking intoaccount these aspects, the current paper shows the following: Given a game withCP, if we additionally assume supermodularity, we can guarantee the existence of aunique 1/2–dominant equilibrium under a generic choice of payoffs. Moreover, giventhat supermodularity is defined on a linearly ordered set of strategies, we can showthat the unique 1/2–dominant equilibrium is either the lowest or the highest strat-egy profile. This implies that, in any generic game with CP and supermodularity,the various equilibrium selection methods support either the lowest or the higheststrategy profile as the prediction of a unique selection among multiple equilibria.

As a connection to the 1/2–dominant equilibrium selection criterion, we con-sider the “potential game method” (Blume, 1995; Monderer and Shapley, 1996) andshow that if a game with CP has a potential function with a unique potential maxi-mizer, the unique potential maximizer is equivalent to a 1/2–dominant equilibrium.This implies that, under CP, the potential game method leads to the same uniqueprediction as the various equilibrium selection methods.

As an application, we consider a two-player case of the “minimum-effort game”defined by Van Huyck et al. (1990). In this game, each player chooses an individualeffort level with a cost. The payoff of each player is equal to two additive separableterms: (i) the benefit depending on the minimum among two players’ effort levels,

1The reason why we do not use the term “total bandwagon property” is that if we do not focuson a context, it is unclear to capture that concept by seeing only the term, while the coordinationproperty seems easier to understand.

2Here, a coordination game is a game where any symmetric pure strategy profile is a strict Nashequilibrium, while a pure coordination game is a coordination game where any symmetric purestrategy profile gives positive payoffs for both players but any other strategy profile gives zero.

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and (ii) the individual effort cost. The distinctive properties of the game are that thebenefit depends on the minimum level of two players’ choices and every symmetricpure strategy profile is a strict Nash equilibrium, each of which is Pareto-ranked.Thus, the minimum-effort game is a coordination game with multiple Pareto-rankedequilibria. In this paper, we would like to see if the minimum-effort game has CP.This is of interest because, given that the minimum-effort game has supermodularity,if CP holds, this paper’s result implies that the various equilibrium selection methodstheoretically support the unique selection of either the lowest or the highest effortlevels among multiple Pareto-ranked equilibria, that is, either the most inefficient orthe most efficient equilibrium is uniquely selected as the theoretical prediction. Weshow that the minimum-effort game has a “limit” version of CP in the sense that ifwe slightly perturb the payoffs of the game, it satisfies CP.

To compare our theoretical prediction with the experimental result in the minimum-effort game, we examine the experimental equilibrium selection result reported byVan Huyck et al. (1990). For the two-player minimum-effort game, they run thetwo types of experiments under the fixed-pair and random-pair repeatedly at tentimes per each session, respectively. Roughly speaking, their experimental resultshows that, under the fixed-pair, almost all subjects select the highest effort leveland achieve the most efficient outcome, whereas under the random-pair, a subjectchooses a low effort but the other chooses a high effort and the effort choices ofthe two subjects do not converge to the unique effort level. For the equilibriumselection result in case of the fixed-pair, Van Huyck et al. (1990) apply the repeatedgame argument to justify the result, which makes sense. For the result in case ofthe random-pair, however, they do not give a justification. This paper provides apotential justification for the unclear result in the random-pair experiment.

Since the minimum-effort game has no 1/2–dominant equilibrium, we cannotapply this paper’s result directly to achieve an integrated theoretical prediction, andso the equilibrium selection methods are unlikely to predict the same selection of theminimum-effort game. This seems to imply that all subjects’ choices are unlikelyto converge to a unique equilibrium. To apply the paper’s result as the integratedtheoretical prediction, we consider a slight perturbations of the minimum-effort gameunder which the perturbed minimum-effort game satisfies CP and supermodularity.Our result implies that the game has a unique 1/2–dominant equilibrium and theunique 1/2–dominant equilibrium is either the most inefficient or the most efficienteffort level profile. Thus, we can provide a strong theoretical prediction for theequilibrium selection in the “perturbed minimum-effort” game.

The remainder of this paper is organized as follows. Section 2 introduces theunderlying symmetric two-player game with CP. Section 3 gives a characterizationof the underlying game with CP. In Section 4, we introduce the equilibrium selectioncriterion, 1/2–dominant equilibrium, and in Section 5, we further assume supermod-ularity and then show several properties of those games. In Section 6, we providea relation between the potential game method and the 1/2–dominance equilibriumselection criterion. In Section 7, we apply the results shown in the previous sec-tions to the minimum-effort game and examine its properties and discuss an issuein the experiment of the minimum-effort game. All proofs except for simple onesare relegated to the appendix.

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2 The Underlying Game

We consider a symmetric two-player game g = (A, g) where A = {1, 2, . . . , n} with#A = n ≥ 2 is the finite set of pure strategies and g : A2 → R is the symmetricpayoff function. If a player and her opponent take strategy i and j, respectively, fori, j ∈ A, the player’s payoff is denoted by gij. We write the set of mixed strategiesby the (n − 1)–dimensional simplex Δ = {x ∈ Rn | ∀i ∈ A, xi ≥ 0,

∑i∈A xi = 1}.

For any x ∈ Δ, let supp(x) = {i ∈ A | xi > 0} be the support of x, and letbr(x) = argmaxi∈A

∑j∈A gijxj and wr(x) = argmini∈A

∑j∈A gijxj be the set of

pure strategy best and worst responses against x, respectively. We introduce aproperty for a class of games, which is equivalent to the “total bandwagon property”introduced by Kandori and Rob (1998) in an economic framework.3

Definition 1. A game g = (A, g) has the coordination property CP if br(x) ⊆supp(x) for any x ∈ Δ.

This condition implies that for each player, if she believes that the opponenttakes some strategies j1, . . . , jk ∈ A for some k = 1, 2, . . . , n, then it is optimal tochoose an strategy among the strategies j1, . . . , jk. Note that any symmetric purestrategy profile is a strict Nash equilibrium because if each player believes that theopponent surely takes a pure strategy j, her optimal choice is the only strategy j.4

In fact, CP is related to the set-valued solution concept, curb set, introduced byBasu and Weibull (1993).5 Since here we focus on symmetric coordination games,we simply introduce the definition of curb sets by using subset of strategies insteadof subset of strategy profiles in the following way.

Definition 2. A subset of strategies S ⊆ A is a curb set if⋃x∈Δ(S)

br(x) ⊆ S(=

⋃x∈Δ(S)

supp(x))

From this definition, it is easy to see that CP is equivalent to the condition thatfor any S ⊆ A, S is a curb set.

As an opposite of CP, we consider the following property, which is introducedby Kojima and Takahashi (2007).

Definition 3. The game g = (A, g) has the anti-coordination property ACP ifwr(x) ⊆ supp(x) for any x ∈ Δ.

Note that the game (A, g) has ACP if and only if the game (A,−g) has CP. Forbrevity, we write −g = (A,−g) subsequently. Kojima and Takahashi (2007) call asymmetric game with ACP an anti-coordination game and provide several propertiesin anti-coordination games as follows.

First, we have:

3Here, we would like to consider the property independently of specific economic situations, andso we do not use the term total bandwagon property.

4Also note that if we take into account dominated strategies, CP does not hold but we canextend all results in this paper straightforwardly because any dominated strategy does not changeall arguments used to show the results.

5For example, see Ritzberger and Weibull (1995) for an investigation of curb set under anevolutionary approach.

4

Fact 1 (Kojima and Takahashi, 2007, Proposition 1). Any anti-coordination gamehas a unique symmetric Nash equilibrium. The unique Nash equilibrium is a com-pletely mixed strategy profile.

A typical example of games with ACP is a hawk-dove game where there existsa unique symmetric Nash equilibrium which is completely mixed over the set ofstrategies.

For any anti-coordination game g, let (x∗,x∗) ∈ Δ2 with x∗ ∈ intΔ be aunique symmetric Nash equilibrium of g which is completely mixed over Δ. Forany nonempty subset S ⊆ A, we denote by g|S the restricted game of g whereevery player chooses a strategy only from the subset of strategies S. Since anyrestricted game g|S preserves the property of anti-coordination game g, g|S has aunique symmetric Nash equilibrium, which is completely mixed over S and denotedby (x∗|S,x∗|S). A characterization of anti-coordination games via restricted gamesis:

Fact 2 (Kojima and Takahashi, 2007, Proposition 5). The game g = (A, g) has ACPif and only if, for any S � A, the restricted game g|S has ACP and wr(x∗|S) = S.

This implies that, to check whether or not a game has ACP, we must check theexistence of a unique interior symmetric Nash equilibrium for any restricted game.

In the next section, we apply Facts 1 and 2 to characterize symmetric two-playercoordination games with CP.

3 Symmetric games with CP

3.1 A Characterization

For g = (A, g), we assume that g has CP. This implies that g is a coordination gameand any symmetric pure strategy profile is a strict Nash equilibrium, that is,

gii > gji, ∀i, j ∈ A. (1)

One of our interests to examine the class of symmetric games with CP is howto know whether or not a symmetric coordination game satisfies CP in an efficientway. As a first step of this interest, we show:

Theorem 1. Let g = (A, g) be a symmetric n × n game. The game g has CP ifand only if the number of symmetric Nash equilibria of g is 2n − 1.

For the proof, see the appendix. We mainly follow the proof of Kojima andTakahashi (2007) and use Facts 1 and 2 given by them. To interpret the aboveresult, we have the following aspects: (i) given any generic symmetric two-playergame, the maximum number of symmetric Nash equilibria is 2n − 1, which is thesame as that of Theorem 1, and so many games may be unlikely to have CP; (ii) theclass of games with CP is the most difficult class of coordination games to choose aunique equilibrium among multiple equilibria because of many equilibrium selectioncandidates.

In the following, we consider two examples that illustrate Theorem 1.

5

1

2

3

1 2 3

1 0 0

0

1

1

00

0

Table 1: A symmetric 3× 3 pure-coordination game

1 2

3

1

3

2

Figure 1: Best response regions of the game of Table 1

Example 1 (Pure coordination game). Let us consider the symmetric 3 × 3 purecoordination game given by Table 1. Note that the class of games with CP includespure coordination games. This implies that the game given by Table 1 satisfies CP.To confirm this, given that any symmetric pure strategy profile of the game is a strictNash equilibrium, we just need to check whether or not there is a (unique) interiorNash equilibrium of the game because the existence of the unique interior Nashequilibrium in addition to all symmetric pure strategy Nash equilibria implies thatany restricted 2 × 2 game has a unique interior Nash equilibrium of the restrictedgame.6 From Figure 1 describing the best response regions of the game, we caneasily see that the game has an interior Nash equilibrium and CP holds. We alsosee that the game has all possible 23 − 1(= 7) symmetric Nash equilibria. Since thegame has CP and 23−1 symmetric Nash equilibria, this is consistent with Theorem1.

Example 2 (Bertrand competition). We consider a Bertrand competition undera discrete set of prices. The Bertrand competition considered here is defined asfollows. There are two symmetric firms 1 and 2 producing a homogeneous goodunder the same cost function. Each firm i = 1, 2 sets price pi from the discrete set{p1, p2, . . . , pn} simultaneously to sell the product to the consumers with the samedemand function.

We define the demand function for consumers and the cost and payoff functionsfor firms in the following. Given the firms’ prices p1 and p2, the consumers buy theproduct from the firm(s) giving the lowest price p = min{p1, p2} between the twofirms, and so their demand for each firm i = 1, 2 against the opponent j is given by

Di(p1, p2) =

⎧⎪⎨⎪⎩d− p if pi < pj ,12(d− p) if pi = pj ,

0 if pi > pj

6Note that this argument holds only for 3 × 3 games. To check why the argument holds onlyfor 3× 3 games, see Honda (2012).

6

where d is a positive constant. Given the firms’ prices and consumers’ demand, thecost function of firm i given (p1, p2) is assumed to be quadratic and given by

C i(Di(p1, p2)) =

⎧⎪⎨⎪⎩c(d− p)2 if pi < pj ,12c(d− p)2 if pi = pj ,

0 if pi > pj

where c is a positive constant.7 Thus, given the firms’ prices p1 and p2, Di(p1, p2)and C i(Di(p1, p2)) for each i = 1, 2, the firm i’s profit is given by

πi(p1, p2) = piDi(p1, p2)− C i(Di(p1, p2)) =

⎧⎪⎨⎪⎩(d− p)(pi − c(d− p)) if pi < pj ,12(d− p)(pi − c

2(d− p)) if pi = pj ,

0 if pi > pj .

Under the above setting, we can see that the Bertrand competition is a price gamebetween two firms.

As a simple example, we assume that the set of price strategies is {1, 2, 3}. Wedefine the Bertrand competition by a symmetric two-player game where two firmsare symmetric players, the set of strategies, A, is {1, 2, 3}, and the payoff functionof each player i = 1, 2 is given by πi : A2 → R. Then, by a simple computation, wecan show that the game is a coordination game if the two parameters c and d satisfysome conditions. Due to the specific payoff function defined above, we can also showthat if the game is a coordination game, the game has CP. On the other hand, ifthe game has CP, the game must be at least a coordination game so that the gamewith CP must satisfy the parameters’ conditions to be a coordination game.

Thus, we have:

Observation 1. We consider the Bertrand competition with A = {1, 2, 3}. Then,CP holds if and only if c ∈ (0, 2/3) and d ∈ (1 + 5c+

√1 + 16c+ 4c2)/3c, 1 + 2/c).

By Theorem 1, we can guarantee that the price game has 23−1 symmetric Nashequilibria if and only if c ∈ (0, 2/3) and d ∈ (1 + 5c+

√1 + 16c+ 4c2)/3c, 1+ 2/c).8

4 Half-dominant equilibrium

In the literature, there are various equilibrium selection methods: (1) the evolu-tionary learning method (Kandori et al., 1993; Young, 1993); (2) the global gamemethod (Carlsson and van Damme, 1993); (3) the incomplete information method(Kajii and Morris, 1997); (4) the perfect foresight dynamics method (Matsui andMatsuyama, 1995); (5) the spatial dominance method (Hofbauer, 1999; Deguchiand Hofbauer, 2010), among others. The common property among those methods is

7As a justification of quadratic cost functions, see Janssen and Karamychev (2010), amongothers. They argue that a convex cost function is relevant to a mobile telephony sector due totelephone line congestions.

8A similar argument is applied for a price game of Varian (1980). For the detail, see Honda(2012).

7

1

2

3

1 2 3

1 0 1

0

3

2

0– 2

0

Table 2: A symmetric 3× 3 coordination game

that if a two-player game has a 1/2–dominant equilibrium, then the selection meth-ods selects the 1/2–dominant equilibrium. The solution concept of 1/2–dominantequilibrium is introduced by Morris et al. (1995) and defined as follows. Note thatthe following definition is based on symmetric coordination game but the formaldefinition is for any two-player game.

Definition 4. A strategy profile (i∗, i∗) ∈ A2 is a 1/2–dominant equilibrium if, forany x ∈ Δ with xi∗ ≥ 1/2,

{i∗} = br(x).

Note that if a strategy profile is a 1/2–dominant equilibrium, then it is a strictNash equilibrium. To make the above condition more clear, we can rewrite it by thefollowing equivalent condition: for strategy profile (i∗, i∗) and any distinct strategyprofile (i, j) ∈ A2 with (i, j) = (i∗, i∗),

1

2gi∗i∗ +

1

2gi∗j >

1

2gii∗ +

1

2gij.

Also, note that if a game has a 1/2–dominant equilibrium, then it is unique.

Example 3. As one of examples for a 1/2–dominant equilibrium, we reconsiderExample 2 of the Bertrand competition with A = {1, 2, 3}. By a simple calcu-lation, we can show that under CP, that is, if c ∈ (0, 2/3) and d ∈ (1 + 5c +√1 + 16c+ 4c2)/3c, 1 + 2/c), the highest price strategy profile, (3, 3), is a (unique)

1/2–dominant equilibrium. This implies that many equilibrium selection methodspredict that the highest price strategy profile is selected among three price strategyprofiles. This result under a discrete set of prices and a quadratic cost function istotally different from the standard Bertrand competition result under a continuousset of prices and a linear cost function where the competition between two firmsleads to the unique equilibrium price equal to the marginal cost.

Note that even under CP, a symmetric coordination game has no 1/2–dominantequilibrium. We can see this by the following example.

Example 4. We consider the symmetric 3× 3 coordination game given by Table 2.The best response regions of the game is given by Figure 2. From Figure 2, we cansee that the game has CP but there is no 1/2–dominant equilibrium.

As another generalization of risk-dominant equilibrium, Kandori and Rob (1998)consider a risk-dominant concept for strategies based on pairwise comparison. Inthe following definition, we consider any symmetric two-player coordination gameand call the strategy in every symmetric Nash equilibrium a Nash strategy.

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1 2

3

1

3

2


Definition 5. Fix two distinct pure strategy Nash equilibria (i, i) and (j, j). Then,Nash strategy i pairwise risk dominates strategy j (i PRD j) if

gii − gji > gjj − gij .

If i PRD j for any Nash strategy j = i, then i is globally pairwise risk dominant(GPRD).

We call a Nash equilibrium in a strategy that is GPRD, a GPRD-equilibrium.Note that if a game has a GPRD-equilibrium, by definition, the GPRD-equilibriumis unique. Also, let us take notice that we have known the following equivalentrelation between GPRD-equilibrium and 1/2–dominant equilibrium under CP.

Fact 3. Let g be a game with CP. Then, a 1/2–dominant equilibrium is equivalentto a GPRD-equilibrium.

Using Fact 3, we can provide the relation between the 1/2–dominant equi-librium and the equilibrium selected by the perfect foresight dynamics method.Roughly speaking, we can argue the relation as follows. On the one hand, Hofbauerand Sorger (2002) show that if a game has a 1/2–dominant equilibrium, the 1/2–dominant equilibrium is selected by the perfect foresight dynamics method. On theother hand, it is easily seen that given a symmetric game with CP, if a symmetricpure strategy profile is selected by the perfect foresight dynamics method, it is aGPRD-equilibrium, which is equivalent to a 1/2–dominant equilibrium by Fact 3.9

Thus, we have:

Fact 4. We consider a symmetric two-player game g = (g, A) with CP. Then, asymmetric pure strategy profile (i, i) is selected by the perfect foresight dynamicsmethod if and only if it is a 1/2–dominant equilibrium.

As in Fact 4, Deguchi and Hofbauer (2010) show that the same relation alsoholds for the spatial dominance method instead of the perfect foresight dynamicsmethod.

5 Supermodular Games

In economics, strategic complementality (supermodularity) has been considered tobe important (Milgrom and Roberts, 1990; Milgrom and Shannon, 1994). Taking

9Note that this holds only for a symmetric game with CP. For the detail, see Hofbauer andSorger (2002).

9

into account this aspect, we first introduce supermodularity and show that any gamewith CP and supermodularity always has a unique 1/2–dominant equilibrium undera generic choice of payoffs, and that the 1/2–dominant equilibrium is either thelowest or the highest strategy profile.

Supermodularity for symmetric n × n games is defined as follows. Let A ={1, 2, . . . , n} be a completely and linearly ordered set of strategies. We assume thatg satisfies supermodularity: for all i, i′, j, j′ ∈ A with i > i′ and j > j′,

gij − gi′j ≥ gij′ − gi′j′. (2)

For supermodular games, let x,x′ ∈ Δ be any two beliefs, and we write x x′ if xstochastically dominates x′, that is, if

∑i≥k xi ≥

∑i≥k x

′i for any k ∈ A with strict

inequality for at least one k. Following Kandori and Rob (1995), we have:

Fact 5 (Kandori and Rob, 1995, Proposition 7). Let g be a supermodular game. Ifx x′, then it follows that

min br(x) ≥ maxbr(x′).

For any symmetric supermodular game with CP, g, by using Fact 5, we showthat g always has a 1/2–dominant equilibrium and further, the unique equilibriumis either the lowest or the highest strategy profile, (1, 1) or (n, n).

To capture the idea of the proof, we consider a symmetric 3 × 3 supermod-ular game with CP and then show that by a generic choice of payoffs, there isalways a unique 1/2–dominant equilibrium, which is either the lowest or the higheststrategy profile, that is, either (1, 1) or (3, 3). To show this, there are two impor-tant points: (i) under a generic choice of payoffs and CP, it follows that for beliefx13 = (1/2, 0, 1/2),

br(x13) = {1} or {3}, (3)

(ii) since the game has supermodularity, we apply Fact 5 to (3) and consider the bestresponse corresponding to two beliefs x12 = (1/2, 1/2, 0) and x23 = (0, 1/2, 1/2). Ifbr(x13) = {1}, since x13 x12,

{1} = min br(x13) ≥ maxbr(x12), (4)

which implies that br(x12) = {1}. On the other hand, if br(x13) = {3}, sincex23 x13,

min br(x23) ≥ maxbr(x13) = {3}, (5)

which implies that br(x23) = {3}. From (4) and (5), it follows that (i) if br(x13) ={1}, br(x12) = {1}; (ii) if br(x13) = {3}, br(x23) = {3}. This shows that either thelowest or the highest strategy profile, (1,1) or (3,3), is a 1/2–dominant equilibrium.This also implies that the intermediate strategy 2 between the lowest and the higheststrategies, 1 and 3, must be pairwise risk-dominated by strategy 1 or 3. Indeed, wecan extend this argument to the games with many strategies.

First, we show:

10

1

2

3

1 2 3

6 5 0

5

8

7

50

5

Table 3: A symmetric 3× 3 game

{1}

{3}

{2}

3

21


Proposition 1. Let g be a generic symmetric n × n supermodular game with CP.Then, g has a unique 1/2–dominant equilibrium. Furthermore, the unique 1/2–dominant equilibrium is either the lowest or the highest strategy profile, (1, 1) or(n, n).

For the proof of Proposition 1, see the appendix.Next, we show the pairwise risk-dominance relationship among strategies. Under

CP, let us consider any restricted game with three strategies and the relationship interms of pairwise risk-dominance among the strategies. Then, we have:

Proposition 2. Given any generic symmetric supermodular game g = (A, g) with#A ≥ 3 and CP, we consider any subset of strategies S ⊆ A with S = {i, j, h}for 1 ≤ i < j < h ≤ n. Let xij and xjh be the beliefs with xij

i = xijj = 1/2 and

xjhj = xjh

h = 1/2, respectively. Then, the restricted game g|S satisfies (i) or (ii):

{(i) br(xij) = {i}(= argminS),(ii) br(xjh) = {h}(= argmaxS).

(6)

This implies that in any restricted game with three strategies, the intermediatestrategy is pairwise risk-dominated by an adjacent strategy of g|S. Note that forany generic supermodular game g with CP, by Proposition 1 and Fact 3, everyintermediate strategy between the lowest and the highest strategies is pairwise risk-dominated by at least one of them.

Example 5. We consider the symmetric 3 × 3 supermodular coordination gamegiven by Table 3. The best response regions of the game are given by Figure 3.Using Figure 3, it is easy to see that the game does not have CP and has no 1/2–dominant equilibrium. Together with Example 4 where in the game CP holds butsupermodularity does not and no 1/2–dominant equilibrium exists, this implies thatCP and supermodularity are indispensable in order to get Proposition 1.

Modifying the payoff matrix in Table 3 by preserving supermodularity, we con-sider the symmetric 3 × 3 supermodular coordination game given by Table 4. The

11

1

2

3

1 2 3

6 3 0

5

8

7

50

2

Table 4: A modified game of Table 3

{1}

{3}

{2}

3

21


best response regions of the game are given by Figure 4. From Figure 4, we cansee that the game satisfies CP. Since the game has both CP and supermodularity,Proposition 1 implies that either (1, 1) or (3, 3) is a unique 1/2–dominant equilibriumof the game. Also, Proposition 2 implies that strategy 2 is pairwise risk-dominatedby strategy 1 or 3. Indeed, from Figure 4, we can easily see that (3, 3) is a unique1/2–dominant equilibrium and strategy 2 is pairwise risk-dominated by strategy 3.

5.1 Computational issue

As one of our interests to examine the class of symmetric games with CP is howto efficiently know whether or not a symmetric coordination game satisfies CP, weconsider an efficient way to see if a game has CP in the following.

If we check whether or not a (symmetric) game has CP, by Theorem 1, we shouldcheck if the number of all symmetric Nash equilibria is 2n−1. Under supermodular-ity, however, we can reduce the computation task by using Proposition 1 and give asimple way to check if a supermodular game has CP. If a supermodular game is nota coordination game, it is obvious that the game does not have CP, and so we con-sider a supermodular coordination game. To this end, we consider a contrapositionof Proposition 1 in the following sense.

Corollary 3. Let g = (A, g) be a generic supermodular coordination game. Ifneither the lowest nor the highest strategy profile is a 1/2–dominant equilibrium,then the game g does not have CP.

This provides a useful and simple way to check whether or not a supermodularcoordination game has CP. For example, let us revisit Example 5, there are twosupermodular coordination games given by Tables 3 and 4. In the game given byTable 3, neither the lowest nor highest strategy profile is a 1/2–dominant equilib-rium, from Corollary 3, we can say that the game does not have CP. On the otherhand, in the game given by Table 4, since (3, 3) is a 1/2–dominant equilibrium, wecan say that the game might have CP.

12

6 Potential Games

We consider a potential game (Monderer and Shapley, 1996) or partnership game(Hofbauer and Sigmund, 1998) in order to show a connection of equilibrium selectionmethods between the potential game method and the other methods introducedin Section 1 given that the game has CP. In the literature, the potential gamemethod has been shown to have several connections among equilibrium selectionmethods including the global game method, the incomplete information method,and the perfect foresight dynamics method (Hofbauer and Sorger, 1999, 2002; Ui,2001; Frankel, Morris and Pauzner, 2003; Morris and Ui, 2005; Oyama, Takahashiand Hofbauer, 2008; Oyama and Tercieux, 2009).10

We follow Monderer and Shapley (1996) for introducing potential games: Givena symmetric game g = (g, A), a function v : A2 → R is a potential function of g iffor any i, i′, j ∈ A, the function v satisfies

vi′j − vij = gi′j − gij.

A pure strategy profile (i∗, j∗) ∈ A is a potential maximizer if there exists a potentialfunction v : A2 → R with (i∗, j∗) ∈ argmax(i,j)∈A2 vij. We call g a potential game ifthere exists a potential function v : A2 → R and a potential maximizer. Note thatany potential maximizer is a Nash equilibrium. Given that the game has CP, sinceany symmetric pure strategy profile is a Nash equilibrium, only a symmetric purestrategy profile can be a potential maximizer.

6.1 Potential maximizer and half-dominant equilibrium

Given a symmetric two-player game with CP, we examine the relation between po-tential maximizer and 1/2–dominant equilibrium. In general, both solution conceptsare independent in the sense that even if one of solutions exists, the other does notnecessarily exist. In this subsection, however, we show that under CP, if a gamehas a unique potential maximizer, the potential maximizer is equivalent to a 1/2–dominant equilibrium.

By the argument of Monderer and Shapley (1996, Corollary 2.9, p.131),11 if agame has a potential function with a potential maximizer, then it follows that forany distinct strategies i, j, i′, j′,

(gji′ − gii′) + (gj′j − gi′j) + (gij′ − gjj′) + (gi′i − gj′i) = 0. (7)

Suppose that a game is a potential game with a potential maximizer. Mondererand Shapley (1996) show that if a game has a potential function v, it is unique in the

10For the relation among the (generilized) potential game method, the global game method, andthe incomplete information method, see Honda (2011), Oyama and Takahashi (2011), and Basteckand Daniels (2011).

11Following Hofbauer and Sigmund (1998, p.84), a game has a potential function if for anyi, j, h ∈ A,

gij + gjh + ghi = gih + ghj + gji.

Actually, this condition is the same as (7) by taking i′ or j′ as i or j.

13

sense that other potential functions are the same as function v by adding constants.For symmetric two-player games, we have:

Fact 6 (Monderer and Shapley, 1996, Lemma 2.7). Fix any game g = (g, A). Let vand v′ be potential functions of g. Then there exists a constant c such that for anystrategy profile (i, j) ∈ A2,

vij − v′ij = c.

This implies that if we find a potential function of a game, it is enough to focuson the potential function in order to apply the potential game method.

By the symmetric payoff function g, since the potential function is common fortwo players, it follows that for any i, j, h ∈ A,

gih − gjh = vih − vjh = vhi − vhj . (8)

From the equality on the right hand side of (8) with h = i, it follows that vij = vjifor any i, j ∈ A. Thus, the potential function is symmetric . By Fact 6 and (8), if asymmetric game has a potential function, then the potential function is a symmetricfunction. Thus, we have:

Fact 7. Fix any symmetric game g = (g, A). Let v be a potential function of g.Then v is symmetric.

From Fact 7, we can show that given that a game with CP has a potentialfunction, then the potential maximizer is equivalent to a 1/2–dominant equilibriumbecause of the following argument. Suppose that a symmetric game has CP whereany pure strategy profile is a Nash equilibrium and there is a potential function vsuch that (i, i) is a unique potential maximizer. Fix any j = i. Since (i) vii > vjj dueto the potential maximizer (i, i) and (ii) vji = vij due to symmetry of the potentialfunction, it follows that

vii − vjj > 0 = vji − vij (by (i) and (ii))

vii − vji > vjj − vij (by exchanging vjj and vji)

gii − gji > gjj − gij (by (8))

1

2(gii − gji) +

1

2(gij − gjj) > 0,

which implies that the potential maximizer is equivalent to a 1/2–dominant equilib-rium.

To sum up the above argument, we have:

Proposition 4. We consider a symmetric two-player game g = (g, A) with CP.Suppose that g has a potential function with a unique potential maximizer. Then,(i, i) is the potential maximizer if and only if it is a 1/2–dominant equilibrium.

To show Proposition 4, we use CP just in order to guarantee the existence of asymmetric Nash equilibrium (i, i). Taking into account this point, more generally,we have:

Corollary 5. We consider a symmetric two-player game g = (g, A). Suppose thatg has a potential function with a unique potential maximizer. Then, (i, i) is thepotential maximizer if and only if it is a 1/2–dominant equilibrium.

14

1

2

3

4

5

6

7

1 2 3 4 5 6 7

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.80

0.70

0.60

0.50

0.40

0.30

0.90

0.80

070

0.60

0.50

1.00

0.90

0.80

0.70

1.10

1.00

0.90

1.20

1.10 1.30

Table 5: The minimum-effort game

7 An application to the minimum-effort game

7.1 The minimum-effort game

Following Van Huyck et al. (1990), we define the minimum-effort game as follows.There are multiple players and we write by I = {1, . . . , I} the set of players. Eachplayer has the finite set of completely and linearly ordered strategies denoted by Awith #A = n. Here, A corresponds to a positive integer set of effort levels. Givenany strategy profile s = (si, s−i), the payoff function of the minimum-effort gamegmini : AI → R for each player i ∈ I is defined as follows.

gmini (si, s−i) = amin{si, s−i} − bsi + c (9)

where a, b, and c are positive constants such that a > b(> 0). From this payofffunction form, the benefit of cooperation depends on the minimum level of effortsamong all players while the cost depends only on the individual effort level. Notethat, independent of the value c(> 0), every pure strategy profile is a Nash equilib-rium.

Specifying the payoff matrix of the minimum-effort games with S = {1, 2, . . . , 7},a = 0.20, b = 0.10, and c = 0.60 as Table 5, Van Huyck et al. (1990) run theexperiments for the two cases of I = 2 and I ∈ {14, 15, 16}. In the case of I = 2,besides the fix-pair experiment, they add the random-pair experiment. For eachsession in each case, they observe the repeated play of the game with 10 rounds andprovide an experimental result for the equilibrium selection. The experimental resultsuggests that in the multiple-player case with I ∈ {14, 15, 16}, the most inefficientequilibrium (1, 1) is selected, while in the two-player case with I = 2 under thefixed-pair, the most efficient equilibrium (7, 7) is selected. In the two-player caseunder the random-pair, however, no equilibrium selection result is not obtained andone player chooses a higher effort level while the other does a lower effort level evenin the final round.

Monderer and Shapley (1996) discuss the experimental result of Van Huyck et al.(1990) via the potential game method. For the payoff function in minimum-effort

15

game given by (9), the potential function is defined by

P (si, s−i) = amin{si, s−i} − b∑j∈I

sj . (10)

We write s = (si, s−i) and the potential maximizer under the potential function (10)is given by

argmaxs∈AI

P (s) =

⎧⎪⎨⎪⎩{(1, . . . , 1)} if a < Ib,

{(n, . . . , n)} if a > Ib,

{(1, . . . , 1), . . . , (n, . . . , n)} if a = Ib.

(11)

From (11), for I–player minimum-effort games with 14 ≤ I ≤ 16 used in the exper-iment of Van Huyck et al. (1990), since a < Ib for a = 0.20 and b = 0.10, there isa unique potential maximizer (1, . . . , 1) which is consistent with the experimentalresult of Van Huyck et al. (1990). On the other hand, since I = 2 for two-playergames and a = Ib = 0.2 in Table 5 used in their experiment, every pure strategyprofile is a potential maximizer, and so we cannot apply the potential game methodto predict the equilibrium selection of the game. One might think that it is quiteunusual for subjects in the experiment to find a potential function and take the strat-egy profile maximizing the potential function. In fact, for the relation between theequilibrium selection of potential games and the experimental result of Van Huycket al. (1990), Monderer and Shapley (1996, p.136, footnote 14) say that “It may bejust a coincidence”.

In the following, we provide the properties of the minimum-effort game in orderto see if we can apply our results in the previous sections to the minimum-effortgame. Also, we give an new insight for the experiment result under the random-pairin Van Huyck et al. (1990) that so far could not be explained.

7.2 Properties of the minimum-effort game

We consider the two properties of the minimum-effort game in the following: theminimum-effort game satisfies (i) supermodularity in terms of effort levels, and (ii)the closure of CP.12

7.2.1 Supermodularity

We can show that the minimum-effort game satisfies supermodularity because forany si, s

′i ∈ A and any s−i, s

′−i ∈ AI−1 with si > s′i and s−i ≥ s′−i,

13

(gi(si, s−i)− gi(s

′i, s−i)

)−(gi(si, s

′−i)− gi(s

′i, s

′−i)

)

= a((min{si, s−i} −min{s′i, s−i})− (min{si, s′−i} −min{s′i, s′−i})

)≥ 0

12In the literature, it is known that the minimum-effort game has supermodularity, but sinceany paper does not mention it as far as I know, it may be useful to show it clearly.

13s−i ≥ s′−i means that for any j = i, sj ≥ s′j .

16

0.70

0.60

0.50

0.40

0.80

0.70

0.60

0.90

0.80 1.00

Table 6: The 4× 4 minimum-effort game

1

2

3

4

x14

x13x24

4

1 2

3 4

4

Figure 5: The best response regions of the game of Table 6

where the inequality follows from the property that the function, min{si, s−i} −min{s′i, s−i}, is weakly increasing in s−i ∈ A,14 and so

min{si, s−i} −min{s′i, s−i} ≥ min{si, s′−i} −min{s′i, s′−i}(≥ 0).

Thus, we have:

Proposition 6. The minimum-effort game is a supermodular game.

This implies that, in the experiment of the minimum-effort game, the higher theother subjects’ effort levels are, the higher each subject’s incentive to increase hisor her effort level is.

7.2.2 Closure of CP

We show that the minimum-effort game satisfies the closure of CP. As the first stepto capture an idea, we consider a symmetric 4×4 minimum-effort game given by thepayoff matrix given by Table 6. The best response regions of the game are given byFigure 5. From Figure 5, we can see that the 4× 4 minimum-effort game “almost”satisfies CP in the sense that br(x) ⊆ supp(x) holds for “almost” all x ∈ Δ exceptfor some points violating CP on the edges over the subsets of strategies, {1, 3},{1, 4}, and {2, 4}. More precisely, over Δ({1, 3}), Δ({1, 4}), and Δ({2, 4}), thereare some mixed strategies x13 ∈ Δ({1, 3}), x14 ∈ Δ({1, 4}), and x24 ∈ Δ({2, 4})such that supp(x13) � br(x13), supp(x14) � br(x14), and supp(x24) � br(x24),respectively, resulting in violations of CP.

From the above observation, we introduce the concept of the closure of CP asfollows.

14See the appendix for the proof of this property.

17

Definition 6. A symmetric game g = (A, g) has the closure of the coordinationproperty (CP) if for any x ∈ Δ, either (i) br(x) ⊆ supp(x) or (ii) supp(x) � br(x)holds.

CP is the “limit” case of CP in the sense that under CP, given any opponent’smixed strategy x ∈ Δ, at least one of best responses for a player is included in thesupport of x, while under CP, all best responses are included in the support. Notethat in general, br(x) ∩ supp(x) = ∅ can hold for some x ∈ Δ.15 Indeed, we canrewrite CP in the following way.

Lemma 1. A game g = (A, g) has CP if and only if for any x ∈ Δ,

br(x) ∩ supp(x) = ∅. (12)

Proof. For the if part, given any x ∈ Δ, br(x) ∩ supp(x) = ∅ implies either (i) or(ii).

For the only if part, since br(x) = ∅ and supp(x) = ∅ for any x ∈ Δ, (i) and (ii)satisfy (12).

Using Lemma 1, we show that the minimum-effort game satisfies CP.

Proposition 7. Let g = (A, g) be the two-player minimum-effort game. Then, ghas CP.

The proof of Proposition 7 is in the appendix. To see that Proposition 7 holds, letus revisit Figure 5 as an example, and then we can see that the 4×4 minimum-effortgame satisfies CP.

7.3 A perturbation of the minimum-effort game

We consider the two-player minimum-effort game. Note that although the gamedoes not satisfy CP, the game does CP by Proposition 7. It is easy to see thatthe minimum-effort game has no 1/2–dominant equilibrium,16 and so we cannotapply this paper’s result directly to achieve an integrated theoretical prediction.This implies that the equilibrium selection methods are unlikely to predict the sameselection in the minimum-effort game, and it seems to imply that all subjects’ choicesare unlikely to converge to a unique equilibrium because if some subjects follow anequilibrium selection method and others follow a different method, the convergenceis not obtained in general.17 To apply the paper’s result as the integrated theoreticalprediction, we consider how to perturb the minimum-effort game in such a way thatCP holds and a 1/2–dominant equilibrium exists. To this end, we consider a convexcombination of the two functions for each player i = 1, 2: (i) the payoff function inthe two-player minimum-effort game, gmin

i , and (ii) a function defined over strategyprofiles, fi : A

2 → R. For any strategy profile (si, sj) and any λ ∈ (0, 1), the convexcombination is given by

gi(si, sj) = λgmini (si, sj) + (1− λ)fi(si, sj). (13)

15As one of examples of games where br(x)∩ supp(x) = ∅ holds for some x ∈ Δ, see Example 5.16It is the same for many-player cases as well.17This is one of criticisms against equilibrium selection methods.

18

From a viewpoint of applications of this paper’s results, in Appendix A, we pro-vide two sufficient conditions to guarantee that the game with the payoff function(13) has CP. Using those sufficient conditions for CP, we can show that the game hasCP if the “perturbed minimum-effort game” with the payoff function (13) satisfiesone of those sufficient conditions. Furthermore, since one of those sufficient condi-tions preserves supermodularity, by Proposition 1, we can show that the perturbedminimum-effort game with satisfying that sufficient condition always has a 1/2–dominant equilibrium, and the 1/2–dominant equilibrium is either the lowest or thehighest (the most inefficient or the most efficient) effort level profile is selected as theintegrated theoretical prediction. For the payoff function (13), however, the functionf is abstract, and so for the experiment, it is desirable to specify the function formbased on some properties in economics. One can find a specific perturbation underwhich the sufficient condition for CP and supermodularity holds.18

Therefore, we have:

Proposition 8. We consider the perturbation of the minimum-effort game via aconvex combination (13). There exist a function fi : A

2 → R for each i = 1, 2 anda parameter λ > 0 such that the perturbed minimum-effort game satisfies CP andsupermodularity.

By Proposition 1, we can further say

Corollary 9. A perturbed minimum-effort game with CP and supermodularity hasa 1/2–dominant equilibrium and the 1/2–dominant equilibrium is either the lowestor the highest strategy profile, that is, either the most inefficient or the most efficientstrategy profile.

From the above, in the perturbed minimum-effort game, either the most ineffi-cient or the most efficient equilibrium is uniquely selected among multiple Pareto-ranked equilibria according to many equilibrium selection methods, while on theother hand, the minimum-effort game has no equilibrium selection criterion stronglysupported by the equilibrium selection methods. It would be interesting to see an ex-perimental result of the perturbed minimum-effort game because we may be able tosee if the theoretical prediction strongly supported by various equilibrium selectionmethods is plausible on the experiment.

8 Conclusion

We have considered the class of games with CP and showed it’s properties, someof which are useful in terms of equilibrium selection. First, we characterized theclass of games with CP via the number of Nash equilibria. Since the number ofsymmetric Nash equilibria is the maximum under a generic choice of payoffs, theequilibrium selection for the class of games with CP is the most difficult in allsymmetric coordination games. Second, we showed that if in addition to CP a gameis supermodular, the game always has a unique 1/2–dominant equilibrium undera generic choice of payoffs, and also showed that the 1/2–dominant equilibrium is

18For an example of such a specific perturbation, see Honda (2012).

19

either the lowest or the highest strategy profile. Given the known result that if agame has a 1/2–dominant equilibrium, various equilibrium selection methods predictthe 1/2–dominant equilibrium as a unique selection among multiple equilibria, thecurrent paper’s result is a useful benchmark as the theoretical prediction.

As an application, we considered the experimental game—the minimum-effortgame, and we examined one of the experimental results of the minimum-effort game.Then, we provided a possible explanation to the experimental result from the theo-retical perspective. To show if this explanation is valuable, or to verify if the strongtheoretical prediction fits with the experimental equilibrium selection, it seems in-teresting to run the experiment for a “simple game” with CP and supermodularityunder which either the lowest or the highest equilibrium is selected by many equi-librium selection methods, such as the perturbed minimum-effort game consideredin Section 7.

Appendix A: Sufficient Conditions for CP

In Appendix A, from a viewpoint of applications of this paper’s results, we givetwo sufficient conditions to guarantee that a game has CP. First, we reconsider thesufficient condition given by Kandori and Rob (1993) and slightly generalize theircondition, which is useful to consider the application of the minimum-effort game.Second, we consider a supermodular game and provide a sufficient condition for CPunder some assumptions. Examples show that these conditions are indeed different.

A.1 The first sufficient condition for CP

Kandori and Rob (1993) consider a completely and linearly ordered set of strategiesand provide a sufficient condition for CP as follows.

Fact 8 (Kandori and Rob, 1993, Proposition 3). Let us consider a symmetric gameg = (A, g). The property CP holds if

gi+1j − gij > gij − gi−1j > 0, ∀i, j ∈ A with i < j,

0 > gi+1j − gij > gij − gi−1j , ∀i, j ∈ A, with i > j.

This condition implies that the payoff function is “one-sided convex” as in Figure6: given any opponent’s pure strategy j ∈ A, the payoff gij for a player takingany pure strategy i ∈ A takes the unique maximum at i = j and decreases as|j− i| is larger, and furthermore, the marginal payoff difference is strictly increasing.Fact 8 is an efficient tool for checking whether or not a symmetric game has CP.Without changing the simplicity to check the condition, we can slightly generalizethe sufficient condition as follows.

Proposition 10. Let us consider a symmetric game g = (A, g). The property CPholds if the following conditions (A.1) and (A.2) hold and at least one of them holdswith strict inequality.

gi+1j − gij ≥ gij − gi−1j ≥ 0, ∀i, j ∈ A with i < j, (A.1)

0 ≥ gi+1j − gij ≥ gij − gi−1j , ∀i, j ∈ A, with i > j. (A.2)

20

gij

ij

Figure 6: An illustration of Kandori and Rob’s sufficient condition

Proposition 10 implies that, even if for any i, j ∈ A with i < j (resp. i, j ∈ A withi > j) condition (A.1) (resp. (A.2)) holds with equality, CP holds under condition(A.2) (resp. (A.1)) with strict inequality. Since the proof is a straightforwardextension of the proof of Kandori and Rob (1993), we omit the proof of Proposition10.

As an opposite condition of the above conditions, we can consider the diminishing-marginal-returns property (DMRP) as in Krishna (1992): given any opponent’s strat-egy j ∈ A,

∀i ∈ A, gi+1j − gij < gij − gi−1j. (A.3)

DMRP imply that given an opponent’s strategy j and two adjacent strategies i, i+1,the payoff difference from the lower strategy to the higher strategy for a player,gi+1j − gij, is decreasing in the player’s own strategy i. In comparison with (A.3),conditions (A.1) and (A.2) correspond to an opposite of DMRP. Note that conditions(A.1) and (A.2) require the sign condition but allow equalities, while DMRP doesnot require the sign condition but requires strict inequalities.

An interesting application of Proposition 10 is the minimum-effort game consid-ered by Van Huyck et al. (1990), because the minimum-effort game satisfies con-ditions (A.1) and (A.2) both with equality. Due to Proposition 10, it can be seenthat a slightly perturbed minimum-effort game in terms of payoffs satisfies CP. Theperturbation includes two types of economic implications: (i) decreasing marginalcost and (ii) positive externality, which are discussed in section 7.

A.2 Supermodular coordination games

To find a sufficient condition for CP, we consider a symmetric supermodular coordi-nation game where any symmetric pure strategy profile is a strict Nash equilibrium,that is, gii > gji for any i, j ∈ A with j = i.

We consider the symmetric 3 × 3 game given by Table 7. The condition ofProposition 10 for this 3 × 3 game is equal to the two conditions: (i) g33 − g23 ≥g23 − g13 ≥ 0 and (ii) 0 ≥ g31 − g21 ≥ g21 − g11 with at least one of them holds withstrict inequalities. So, we can easily see that the game satisfies the two conditions.Thus, the game given by Table 7 has CP.

In comparison with Proposition 10, we derive a different sufficient condition for

21

1

2

3

1 2 3

6 0 0

3

8

7

50

2

Table 7: A symmetric 3× 3 coordination game

1

2

3

1 2 3

6 2 0

5

8

7

50

2

Table 8: A modified game of Table 7

CP. To capture the intution of it, we consider the following example where thecondition of Proposition 10 does not hold but CP holds under supermodularity.

Example 6. We consider the symmetric 3 × 3 coordination game given by Table8, which modifies the payoff matrix of Table 7. It is easy to see that the gamesatisfies supermodularity. Since the condition of Proposition 10 is equal to the twoconditions (i) and (ii) given by the above argument, one can see that condition (i)does not hold. The best response regions of the game are given by Figure 7. It iseasy to see that the game has CP.

It is possible to extend the argument used in Example 6 to a class of games withmany strategies. In fact, for a symmetric n × n supermodular coordination game,we provide a sufficient condition for CP below.

A.3 The second sufficient condition for CP

We consider a symmetric n × n supermodular coordination game. To guaranteethat the symmetric n × n supermodular coordination game has CP, we introducetwo properties and give a sufficient condition for CP under those properties.

The first property is a weaker version of CP.

Definition 7. A game g = (A, g) has the weak coordination property (wCP) if, forany two strategies i, j ∈ A with i < j which are not adjacent, any strategy k inS = {i+ 1, · · · , j − 1}, and any belief x ∈ Δ({i, j}),

k /∈ br(x). (A.4)

{1}

{3}

{2}

3

21


22

The wCP implies that an intermediate strategy k between some two distinctstrategies i and j cannot be a best response for any belief in which the opponenttakes the two strategies i and j. Equivalently, each player is better to take a loweror higher strategy than the intermediate strategies in S.

As a bit stronger property than wCP, Maruta (1997) introduces the “pairwisebandwagon property” (PBP).19 In fact, PBP requires that

∀i, j ∈ A with i = j, br(x) ⊆ supp(x), ∀x ∈ Δ({i, j}). (A.5)

Clearly, wCP is weaker than PBP because of the following. Given a game g, supposethat there is a strategy h which is lower than i or higher than j. On the one hand,under wCP, it can be the case that strategy h is a best response for some beliefx ∈ Δ({i, j}), while on the other hand, PBP rules out the case. Note that thecondition of Proposition 10 implies wCP but not vice versa.

To achieve a sufficient condition for CP in a supermodular coordination game,we further need to consider the following assumption. Suppose that there are twonon-adjacent strategies i, j with i < j. Then, we consider any lower strategy lthan i and any higher strategy h than j if any. For some intermediate strategy kwith i < k < j, we consider payoff differences gim − gkm and gkm − gjm given theopponent’s strategy m ∈ A, and we define the relative payoff difference given theopponent’s strategy m by

Δi,k,jm ≡ gim − gkm

gkm − gjm.

Using the relative payoff difference, we introduce the following property that relativepayoff differences are monotone in opponent’s strategies.

Definition 8. A game g = (A, g) satisfies monotone relative payoff difference(MRPD) if, for any two strategies i, j ∈ A with i < j which are not adjacent,any strategy k in S = {i + 1, · · · , j − 1}, and any strategy m ∈ A\S, the relativepayoff difference Δi,k,j

m is weakly decreasing in the opponent’s strategy m.

Note that, given non-adjacent two strategies i, j and S = {i + 1, · · · , j − 1},MPRD holds only for the opponent’s strategies out of S. Using wCP and MPRD,we provide a sufficient condition for CP as follows.

Proposition 11. A symmetric n × n supermodular coordination game has CP ifthe game has both wCP and MRPD.

The proof of Proposition 11 is relegated to the appendix. The key point of theproof is that a supermodular game with wCP and MRPD has a dominant mixedstrategy consisting of specific two strategies in the support against any intermediatestrategy outside the support. In this sense, the proof is similar to that of Proposition10. In Proposition 11, technically we can replace MRPD with the following weakercondition to guarantee CP.

19As an another weaker property, Alos-Ferrer and Weidenholzer (2007) introduce “partial band-wagon property”.

23

15

12

14

18

11

10

6 7

5 5 13

13

910

7 71

2

3

4

1 2 3 4

Table 9: A 4× 4 supermodular coordination game

Definition 9. A game g = (A, g) satisfies weak monotone relative payoff difference(wMRPD) if, for any two strategies i, j ∈ A with i < j which are not adjacent, anystrategy k in S = {i+ 1, · · · , j − 1}, and any lower (resp. higher) strategy l (resp.h) with l < i (resp. j < h),

Δi,k,jl ≥ Δi,k,j

i , Δi,k,jj ≥ Δi,k,j

h .

The difference between MRPD and wMRPD is the following. Although therelative payoff difference Δi,k,j

m is weakly decreasing in the opponent’s strategy m ∈A\S under MRPD, Δi,k,j

m can be increasing but is constrained by Δi,k,ji or Δi,k,j

j

under wMRPD.

Corollary 12. A symmetric n × n supermodular coordination game has CP if thegame has both wCP and wMRPD.

A.4 Examples of sufficient conditions for CP

In the following two examples, we show that the conditions in Proposition 10 areindependent of those in Proposition 11. The first example is a game where theconditions in Proposition 10 are satisfied but the ones in in Proposition 11 are not.The second example is the opposite. Note that both games in the examples haveCP because the games satisfy one of the sufficient conditions for CP.

Example 7. We consider a symmetric 4 × 4 coordination game given by Table 9.The best response regions of the game are given by Figure 9 where each black pointcorresponds to a pure strategy Nash equilibrium, each blue point a completely mixedstrategy Nash equilibria over S with #S = 2, each red point a completely mixedstrategy Nash equilibria over S with #S = 3, and the green point a unique interiorNash equilibrium. From Table 9, one can easily see that the game satisfies the

1

2

3

4


24

15

12

14

18

11

11

6 7

5 6 13

16

1311

10 121

2

3

4

1 2 3 4

Table 10: A 4× 4 supermodular coordination game

1

2

3

4


conditions of Proposition 10. Therefore, Proposition 11 guarantees that the gamehas CP. Actually, from Figure 8, it is easy to see that the game has CP. Moreover,from Table 9, we can see that the highest strategy profile (4, 4) is a 1/2–dominantequilibrium.

Note that the game violates supermodularity because g23 − g13 > g24 − g14,g31 − g21 > g32 − g22, and g41 − g31 > g42 − g32. Also let us take notice that since a4 × 4 game with CP must have 24 − 1 Nash equilibria according to Theorem 1, itis easy to see that the game actually has 15 Nash equilibria. All (symmetric) Nashequilibria x ∈ Δ are:

• 4 Pure strategy Nash equilibria: x = ei for any i ∈ A.

• 6 completely mixed strategy Nash equilibria over S with #S = 2:

(1

6,5

6, 0, 0), (

7

16, 0,

9

16, 0), (

11

21, 0, 0,

10

21), (0,

4

9,5

9, 0), (0,

9

16, 0,

7

16), (0, 0,

5

6,1

6)


(19

56,1

7,29

56, 0), (

23

98,5

14, 0,

20

49), (

41

96, 0,

13

32,1

6), (0,

4

9,17

54,13

54).

• A unique interior Nash equilibrium: (101/328, 7/41, 107/328, 8/41).

Example 8. We consider a symmetric 4 × 4 coordination game given by Table10. The best response regions of the game are given by Figure 9 where each blackpoint corresponds to a pure strategy Nash equilibrium, each blue point a completelymixed strategy Nash equilibria over S with #S = 2, each red point a completelymixed strategy Nash equilibria over S with #S = 3, and the green point a uniqueinterior Nash equilibrium. From Figure 9, we can easily see that the game has

25

wCP, and from Table 10, one can easily see that the game has MPRD. Therefore,Proposition 11 guarantees that the game has CP. Actually, from Figure 9, it is easyto see that the game has CP. Since the game has CP and satisfies supermodularity,we can apply Proposition 1 that the game always has a 1/2–dominant equilibrium,and the 1/2–dominant equilibrium is either lowest or highest strategy profile. Thisimplies that to find a 1/2–dominant equilibrium of the game, it is enough to checkthe pairwise risk dominant strategy only over the subset of strategies, {1, n}. FromTable 10, since strategy 1 pairwise risk dominates strategy n, by Proposition 11, thelowest strategy profile (1, 1) is a 1/2–dominant equilibrium.

Note that the game does not satisfy the conditions (A.1) and (A.2) of Proposition10 because g31−g21 < g21−g11 and g44−g34 < g34−g24. Also let us take notice thatsince a 4× 4 game with CP must have 24 − 1 Nash equilibria according to Theorem1, it is easy to see that the game actually has 15 Nash equilibria. All (symmetric)Nash equilibria x ∈ Δ are:

• 4 Pure strategy Nash equilibria: x = ei for any i ∈ A.


(1

5,4

5, 0, 0), (

4

13, 0,

9

13, 0), (

3

8, 0, 0,

5

8), (0,

3

8,5

8, 0), (0,

5

11, 0,

6

11), (0, 0,

2

3,1

3)


(1

5,7

40,5

8, 0), (

1

5,14

55, 0,

6

11), (

4

13, 0,

14

39,1

3), (0,

3

8,7

24,1

3).

• A unique interior Nash equilibrium: (1/5, 7/40, 7/24, 1/3).

Appendix B: Proofs

In Appendix B, we provide all proofs that are not given.

Proof of Theorem 1

To characterize symmetric two-player games with CP via the number of Nash equi-libria, we mainly follow the proof of Kojima and Takahashi (2007) and use Facts 1and 2 given by them.

Proof. We show the if part as follows. If the number of symmetric Nash equilibriain symmetric game g = (A, g) with #A = n is 2n − 1 then the game must have allpossible symmetric Nash equilibria:

{x∗ ∈ Δ | x∗i > 0, x∗

j = 0, ∀i, j ∈ A, i = j,

n∑i=1

x∗i = 1,

#supp(x∗) = k, ∀k = 1, . . . , n}. (A.6)

26

From (A.6) with k = 1, all symmetric pure strategy profiles are Nash equilibria,and so br(x) ⊆ supp(x) holds for the set of beliefs denoted by X1 = {x ∈ Δ |#supp(x) = 1}.

To show that CP is satisfied, we have to show that br(x) ⊆ supp(x) also holds forthe set of beliefs X2 ≡ Δ\X1. For any belief x ∈ intΔ, br(x) ⊆ supp(x) obviouslyholds. So, we consider the other beliefs x ∈ X2\intΔ. For any x ∈ X2\intΔ, wereconstruct x via a sequence of symmetric Nash equilibria of restricted games asfollows. Using the minimum ratio rule, we consider a sequence of pairs of restrictedset of strategies and some constants, {(Sk, ck)}k with Sk ⊆ A and ck > 0 fork = 0, 1, . . . , m such that

S0 = supp(x), c0 = mini∈S0

xi

x∗i |S0

=xi0

x∗i0|S0

, (i0 = argmini∈S0

xi

x∗i |S0

),

S1 = S0\ argmini∈S0

xi

x∗i |S0

= S0\{i0},

c1 = mini∈S1

xi − c0x∗i |S0

x∗i |S1

=xi1 − c0x∗

i1|S0

x∗i1|S1

, (i1 = argmini∈S1

xi − c0x∗i |S0

x∗i |S1

),

(1 ≤ k ≤ m) Sk+1 = Sk\ argmini∈Sk

xi −∑

j<k cjx∗

i |Sj

x∗i |Sk

= Sk\{i0, i1, . . . , ik},

ck+1 = mini∈Sk+1

xi −∑

j<k+1 cjx∗

i |Sj

x∗i |Sk+1

,

∅ = Sm+1 � Sm � · · · � S1 � S0.

Using the above sequence {(Sk, ck)}k, we can write x by

x =m∑

k=0

ckx∗|Sk .

By the minimum ratio rule, one can show that

br(x) =

m⋂k=0

br(x∗|Sk). (A.7)

Since br(x∗|Sk) = supp(x∗|Sk) = Sk and Sk+1 � Sk for k = 0, 1 . . . , m− 1, togetherwith (A.7), we have

br(x) =m⋂k=0

br(x∗|Sk) =m⋂k=0

Sk = Sm � S0 = supp(x).

Note that if m = 0, br(x) = supp(x) holds. This implies that br(x) ⊆ supp(x)holds for any x ∈ X2\intΔ.

Therefore, since we have shown that br(x) ⊆ supp(x) holds for any x ∈ Δ, CPis satisfied.

Next, we show the only if part as follows. From Fact 1, for any symmetric game−g = (A,−g) with ACP, any restricted game −g|S for S ⊆ A has a unique Nashequilibrium, x∗|S, which is completely mixed over S. Since, for any S ⊆ A, −g|S

27

satisfies ACP if and only if g|S satisfies CP, this implies that the game g|S with CPhas a unique Nash equilibrium which is completely mixed over S for any S ⊆ A.This is because of the following argument. From Fact 2(: wr(x∗|S) = supp(x∗|S) forany S ⊆ A) and the property of Nash equilibria(: supp(x∗|S) ⊆ br(x∗|S)), it followsthat for any x∗|S,

wr(x∗|S) = supp(x∗|S) = br(x∗|S). (A.8)

From (A.8), for any S ⊆ A,

S = argmini∈A∑j∈A

(−gij)x∗j |S (by S = supp(x∗|S) = wr(x∗|S))

= argmaxi∈A∑j∈A

gijx∗j |S.

For all k ∈ S and h ∈ A\S, ∑j∈A

gkjx∗j |S >

∑j∈A

ghjx∗j |S.

Hence, for any S ⊆ A, the game g with CP has a Nash equilibrium x∗|S which iscompletely mixed over S.

To complete the proof, we must show the uniqueness of a Nash equilibriumx∗|S which is completely mixed over S for any S ⊆ A. Assume that there aretwo distinct Nash equilibria, x∗|S and x∗∗|S, which are both completely mixed overS. Since any linear combination of x∗|S and x∗∗|S over Δ(S) must be a Nashequilibrium, a linear combination x of the two Nash equilibria on the boundary ofΔ(S) is also a Nash equilibrium which is completely mixed over bdΔ(S).20 Notethat br(x) = S.21 This is a contradiction to CP because the Nash equilibrium xmust satisfy br(x) ⊆ supp(x) due to CP, but supp(x) = bdΔ(S) � S, resulting inbr(x) � S.

Thus, any symmetric game g with CP has a unique Nash equilibrium which iscompletely mixed over S for any S ⊆ A. This implies that the number of Nashequilibria of g is equal to the number of possible subsets of strategies, that is,2n − 1.

An alternative proof of the only if part of Theorem 1

For the only if part of Theorem 1, we provide an alternative proof based on theoddness property of Nash equilibria in the following.

Let us consider any symmetric two-player game g = (A, g) with CP and #A ≥ 2.First, by induction, we show:

Lemma 2. Let g = (A, g) be any symmetric two-player game with CP and #A ≥ 2.Then any restricted game g|S with #S ≤ k(= 2, 3, . . . , n) has a unique symmetricNash equilibrium which is completely mixed over S.

20For the details, see Honda (2012).21The reason why br(x) = S holds is that x is a linear combination of the two Nash equilibria,

x∗|S and x∗∗|S , with br( ˜x∗|S) = br( ˜x∗∗|S) = S.

28

For the proof, let us take notice that if g has CP then any restricted game g|Shas CP as well.

Proof. We show Lemma 2 by induction as follows.

(I) It is obvious for k = 2.

(II) Suppose that any restricted game g|S with #S ≤ k(= 2, 3, . . . , n − 1) has aunique symmetric Nash equilibrium which is completely mixed over S. Thisimplies that any restricted game g|S with #S = k′ ≤ k has 2k

′ − 1 symmetricNash equilibria for any k′ = 1, 2, . . . , k. Fix the restricted game g|S′ with#S ′ = k + 1. Given S ′, we consider all combinations of the restricted gameswith at least one strategy that we can construct from S ′. The number ofthe combinations of the restricted games except for g|S′ is 2k+1 − 2 and thecorresponding restricted game to each combination has a unique symmetricNash equilibrium. Thus, g|S′ has at least 2k+1 − 2 symmetric Nash equilibria.If g|S′ has more symmetric Nash equilibria than 2k+1 − 2, the only possibilityis to have a unique symmetric Nash equilibrium which is completely mixedover S. By the oddness property of Nash equilibria, since the number of Nashequilibria 2k+1 − 2 is even, g|S′ must have the unique Nash equilibrium.22

Using Lemma 2, we show only if part of Theorem 1.

Proof. From Lemma 2, since any restricted game g|S with #S ≤ n−1 has a uniquesymmetric Nash equilibrium which is completely mixed over S and the number ofthe restricted games g|S with #S ≤ n− 1 is 2n − 2, the game g|S as #S = n, thatis, g = (A, g) has at least 2n − 2 symmetric Nash equilibria and furthermore, byLemma 2, the game must have a unique interior Nash equilibrium. Thus, g = (A, g)has (2n − 2) + 1 = 2n − 1 symmetric Nash equilibria.

Proof of Proposition 1

Proof. For the belief x1n ∈ Δ with x1n1 = x1n

n = 1/2 and x1nj = 0 for any j ∈

A\{1, n}, CP implies that

br(x1n) ⊆ {1, n}.By a generic choice of payoffs, br(x1n) = {1} or {n}. Suppose that br(x1n) = {1}.For any k ∈ A\{1}, let x1k ∈ X be any belief with x1k

1 = 1/2 and x1kk = 1/2. Since

x1n x1k for any k ∈ A\{n}, from Fact 5, it follows that

{1} = min br(x1n) ≥ maxbr(x1k), (A.9)

which implies that for any k ∈ A and any belief x1k ∈ X,

br(x1k) = {1}. (A.10)

22Note that for any symmetric game with CP, there is no asymmetric Nash equilibrium of thegame.

29

The condition (A.10) implies that strategy 1 pairwise risk-dominates any distinctstrategy k for any k ∈ A\{1}, that is, strategy 1 is a GPRD-equilibrium. Thus, byFact 3, strategy 1 is a 1/2–dominant strategy.

To show that (1, 1) is a unique 1/2–dominant equilibrium, suppose that thereare two 1/2–dominant equilibria, (i, i) and (j, j). This gives the following two in-equalities,

1

2g(i, i) +

1

2g(i, j) >

1

2g(j, j) +

1

2g(j, i),

1

2g(j, j) +

1

2g(j, i) >

1

2g(i, i) +

1

2g(i, j),

a contradiction.If br(x1n) = {n} then we can similarly show that (n, n) is a unique 1/2–dominant

equilibrium.


Proof. To prove by a contradiction, assume that

br(xij) = {j} and br(xjh) = {j} (A.11)

where xij and xjh are the beliefs with xiji = xij

j = 1/2 and xjhj = xjh

h = 1/2. For

i < j < h and the belief xih with xihi = xih

h = 1/2, xih xij and xjh xih. Byapplying Fact 5 and (A.11) to xih xij and xjh xih,

min br(xih) ≥ maxbr(xij) = {j} = min br(xjh) ≥ maxbr(xih),

resulting in

br(xih) = {j} /∈ supp(xih),

a contradiction to CP.


To show Proposition 6, by the argument in the subsection 7.2, we just need to showthat for any si, s

′i, s−i ∈ A with si > s′i, the function min{si, s−i} − min{s′i, s−i} is

weakly increasing in s−i ∈ A.

Proof. Take any two distinct actions s−i, s′−i ∈ A with s−i > s′−i. We show that for

any si, s′i ∈ A with si > s′i, min{si, s−i}−min{s′i, s−i} ≥ min{si, , s′−i}−min{s′i, s′−i}.

To this aim, we consider six possible cases for any si, s′i, s−i, s

′−i ∈ A with si > s′i

and s−i > s′−i as follows.

Case 1: si > s′i ≥ min{s−i}(> min{s′−i}).min{si, s−i} −min{s′i, s−i} = min{s−i} −min{s−i}

= 0 = min{s′−i} −min{s′−i}= min{si, s′−i} −min{s′i, s′−i}.

30

Case 2: si ≥ min{s−i} > s′i ≥ min{s′−i}.min{si, s−i} −min{s′i, s−i} = min{s−i} − s′i

> 0 = min{si, s′−i} −min{s′i, s′−i}.

Case 3: si ≥ min{s−i} > min{s′−i} > s′i.

min{si, s−i} −min{s′i, s−i} = min{s−i} − s′i> min{s′−i} − s′i= min{si, s′−i} −min{s′i, s′−i}.

Case 4: min{s−i} > si ≥ min{s′−i} > s′i.

min{si, s−i} −min{s′i, s−i} = si − s′i≥ min{s′−i} − s′i= min{si, s′−i} −min{s′i, s′−i}.

Case 5: min{s−i} > si > s′i ≥ min{s′−i}.min{si, s−i} −min{s′i, s−i} = si − s′i

> 0 = min{s′−i} −min{s′−i}= min{si, s′−i} −min{s′i, s′−i}.

Case 6: min{s−i} > min{s′−i} > si > s′i.

min{si, s−i} −min{s′i, s−i} = si − s′i= min{si, s′−i} −min{s′i, s′−i}.

From Case 1 to Case 6, it follows that

min{si, s−i} −min{s′i, s−i} ≥ min{si, , s′−i} −min{s′i, s′−i}.Thus, we have shown that for any si, s

′i, s−i ∈ A with si > s′i, the function

min{si, s−i} −min{s′i, s−i} is weakly increasing in s−i ∈ A.


First of all, we provide one lemma to show Proposition 7. Then, using that lemma,we prove Proposition 7.

We show:

Lemma 3. Fix any S ⊆ A with #S = 2 and any x ∈ Δ(S). Let S = {l, h} forl, h ∈ A with l < h. Then, if k ∈ A with l < k < h exists, then there exists a uniquepoint x∗ ∈ intΔ({l, h}) with x∗

l = (a− b)/a and x∗h = 1− x∗

l such that

supp(x∗) � br(x∗), (A.12)

br(x) ⊆ supp(x), ∀x ∈ Δ({l, h})\{x∗}. (A.13)

31

Proof. Fix any S ⊆ A with #S = 2, which is denoted by S = {l, h} with l < h.Note that since the minimum effot game is a coordination game, br(ei) = {i} fori = l, h. For any x ∈ Δ(S) with xh = 1− xl and any m ∈ A with l < m ≤ h,

n∑j=1

gljxj −n∑

j=1

gmjxj =∑j∈S

gljxj −∑j∈S

gmjxj

=(xl(amin{l, l} − bl + c) + (1− xl)(amin{l, h} − bl + c)

)

−(xl(amin{m, l} − bm+ c) + (1− xl)(amin{m, h} − bm+ c)

)

=(xl(a− b)l + (1− xl)(a− b)l

)−(xl(al − bm) + (1− xl)(a− b)m

)

=(a− b)l −(xl(al − bm) + (1− xl)(a− b)m

)=(a− b)(l −m)− a(l −m)xl

=(m− l)(axl − (a− b)

). (A.14)

For (A.14), since m− l > 0,

n∑j=1

gljxj −n∑

j=1

gmjxj

⎧⎪⎨⎪⎩> 0 if xl > x∗

l = (a− b)/a,

= 0 if xl = x∗l ,

< 0 if xl < x∗l .

(A.15)

From (A.14) and (A.15), for any x ∈ Δ(S),

argmaxl≤m≤h

( n∑j=1

gljxj −n∑

j=1

gmjxj

)⎧⎪⎨⎪⎩= {l} if xl > x∗

l ,

= {m ∈ A | l ≤ m ≤ h} if xl = x∗l ,

= {h} if xl < x∗l .

(A.16)

Therefore, we have shown that any intermediate pure strategy between l and hcannot be a best response for any x ∈ Δ(S) except for a unique point x∗

To show Lemma 3, it remains to show that any other pure strategy j ∈ A witheither j < l or j > h (if any) cannot be a best response for any x ∈ Δ(S) as well.Since the minimum-effort game is a coordination game and also a supermodulargame by Proposition 6, strategy j is strictly dominated by strategy l or h.23 Thisimplies that strategy j cannot be a best response.

Thus, for any S = {l, h} for l, h ∈ A with l < h, (A.12) and (A.13) hold.

Next, using Lemma 3, we show Proposition 7.

Proof. If x ∈ intΔ, since supp(x) = A, the condition (12) obviously holds. So, weconsider x ∈ Δ\intΔ in the following.

23For the detail, see the conditions (A.20) and (A.21) used in the proof of Proposition 11.

32

Fix any S ⊆ A with #S = 1, . . . , n − 1 and any x ∈ Δ(S). For subset ofstrategies S, we consider two possible cases as follows. The proof is to show that forany strategy k ∈ A\S = ∅, there is a weakly dominant strategy given any opponent’sstrategy x.

Case 1: Either k < i or k > i for any i ∈ S.In this case, since strategy k is strictly dominated by some action j ∈ S,24

strategy k cannot be a best response, that is, k /∈ br(x) for any x ∈ Δ(S).

Case 2: l < k < h for the lowest and highest strategies l, h ∈ S.In this case, there are two strategies i, j ∈ S such that i < k < j with {i} =argmin{s ∈ S | k − s > 0} and {j} = argmax{s ∈ S | k − s < 0}. By Lemma3, for any x ∈ Δ({i, j}), we know that k cannot be a best response except fora unique point denoted by x|∗{i,j}. Next, we consider any opponent’s mixed

strategy x ∈ Δ({m,m′}) with either l ≤ m ≤ i and j ≤ m′ ≤ h. By a specificpayoff function in the minimum-effort game, we can apply the argument ofLemma 3. This means that for any x ∈ Δ({m,m′}) and x∗ ∈ Δ({m,m′})such that x∗

m = (a − b)/a and x∗m′ = 1 − x∗

m, it follows that k /∈ br(x) forany x ∈ Δ({m,m′})\{x∗} and k ∈ br(x∗) with supp(x∗) � br(x∗). Since anyx ∈ Δ(S) can be expressed by a weighted average of x ∈ Δ({m,m′}) for alll ≤ m ≤ i and j ≤ m′ ≤ h, we can show that for any x ∈ Δ(S) and any k,k /∈ br(x) or k ∈ br(x), and if k ∈ br(x), supp(x) � br(x).

From Cases 1 and 2, we have shown that for any x ∈ Δ(S) and any k ∈ A\S,k /∈ br(x) or k ∈ br(x), and if k ∈ br(x), the only possible case is supp(x) � br(x).

Therefore, we have shown that the minimum-effort game satisfies CP.


To show Proposition 11, we provide several properties of supermodular coordinationgames under the condition (A.4) as follows.

Let us consider a symmetric n× n supermodular coordination game g where forany two strategies i, j ∈ A with i = j,

gii > gji. (A.17)

First, by supermodularity and (A.17), one can show:

Lemma 4. Fix any two distinct strategies l, h ∈ A with l < h. It follows that

∀i, j ∈ A with i < j ≤ h and ∀k ≥ h, gjk − gik > 0, (A.18)

∀i, j ∈ A with l ≤ i < j and ∀k ≤ l, gjk − gik < 0. (A.19)

To capture an intuition behind conditions (A.18) and (A.19), we consider thefollowing. We write two subsets of strategies by S � A and S ′ � S where thelowest and highest strategies are l, h ∈ S with l < h and S ′ = S\{l, h}. Letus consider a situation where both two players believe that the opponent takes a

24For the detail, see the conditions (A.20) and (A.21) used in the proof of Proposition 11.

33

l

h

l h

R1

R2

R3

Figure 10: An illustration of the conditions (A.18) and (A.19)

l

h

l h

R1

R2

R3

R4

R5

Figure 11: An illustration of the conditions (A.20) and (A.21)

strategy k ∈ A\S ′. For this situation, we consider the payoff matrix for each playerand some better response relationships among strategies under the beliefs that theopponent takes strategy k. See Figure 10 where the region R1 represents the payoffmatrix under the restricted strategy space S, the region R2 the payoff matrix forthe restricted strategy profiles in which a player and the opponent take strategiess ≤ h and k ≥ h, respectively, and the region R3 the payoff matrix for the restrictedstrategy profiles in which a player and the opponent take strategies s ≥ l and k ≤ l,respectively. In Figure 10, the condition (A.18) means that in region R2 where theplayer believes that the opponent takes a strategy k higher or equal to h, it is betterfor the player to take a higher strategy j ≤ h than a lower strategy i < j, whilethe condition (A.19) means that in region R3 where the player believes that theopponent takes a strategy k lower or equal to l, it is better for the player to take aa lower strategy i ≥ l than a higher strategy j > i.

Note that for Lemma 4, we can say an additional property that

∀i, j ∈ A with i < j ≤ l and ∀l ≤ k ≤ h, gjk − gik > 0, (A.20)

∀i, j ∈ A with h ≤ i < j and ∀l ≤ k ≤ h, gjk − gik < 0. (A.21)

As an illustration of the conditions (A.20) and (A.21), see Figure 11 where thecondition (A.20) means that in the regions R4 where the player believes that theopponent takes a strategy k between l and h, it is better for the player to take ahigher strategy j ≤ l than a lower strategy i < j, while the condition (A.21) meansthat in region R5 where the player believes that the opponent takes a strategy kbetween l and h, it is better for the player to take a a lower strategy i ≥ h than ahigher strategy j > i.

Using Lemma 4 and the conditions (A.20) and (A.21), we can show:

Lemma 5. Fix any opponent’s mixed strategy x ∈ Δ\intΔ. Then, for any purestrategy out of the support of x, k ∈ A\supp(x), there exists a strictly dominantstrategy against k given the opponent’s strategy x.

34

To show Lemma 5, we consider the two possible cases for x ∈ Δ\intΔ.25 Tomake the following notations simpler, given x ∈ Δ\intΔ, we write the minimumand maximum of supp(x) by l and h, respectively.

Case (i): x ∈ Δ\intΔ such that �k ∈ A\supp(x) with l < k < h.By (A.20) and (A.21), since either k < l or k > h, either l or h is a strictdominant strategy against strategy k.

Case (ii): x ∈ Δ\intΔ such that ∃k ∈ A\supp(x) with l < k < h.For any k < l or k > h, by the same argument in Case (i), strategy k cannot bea best response. So, we consider any strategy k with l < k < h and fix k. Letthe pure strategies l′ and h′ in supp(x) be such that {l′} ≡ argmins∈supp(x){k−s | s < k} and {h′} ≡ argmaxs∈supp(x){k−s | s > k}. Note that if there existspure strategy s with l′ < s < h′, s /∈ supp(x). We show that for any opponent’smixed strategy x ∈ Δ\intΔ in Case (ii), strategy k is strictly dominated bysome strategy x′ ∈ Δ({l′, h′}). By wCP, for the two strategies l′ and h′ andany opponent’s strategy x ∈ Δ({l′, h′}), strategy k cannot be a best responseagainst x′. From this, we just need to consider the opponent’s mixed strategyx including pure strategies which are lower than or equal to l′ or higher thanor equal to h′, that is, pure strategies m with l < m ≤ l′ or strategy m withh′ ≤ m < h. By MRPD, it follows that

∀m ∈ {l + 1, . . . , l′}, gl′m − gkmgkm − gh′m

≥ gl′l′ − gkl′

gkl′ − gh′l′, (A.22)

∀m ∈ {h′, . . . , h− 1}, gl′h′ − gkh′

gkh′ − gh′h′≥ gl′m − gkm

gkm − gh′m. (A.23)

For (A.23), since each numerator and denominator on both sides are negative,we rewrite (A.23) as follows.

∀m ∈ {h′, . . . , h− 1}, gh′m − gkmgkm − gl′m

≥ gh′h′ − gkh′

gkh′ − gl′h′. (A.24)

Without loss of generality, we normalize the payoffs gkm for all m ∈ {l +1, . . . , l′} or m ∈ {h′, . . . , h − 1} to be zero and the positive denominators ofboth conditions (A.22) and (A.24) to be 1, and we parameterize the numeratorsof them via positive parameters, tm, tl′, tm, and th′. And we rewrite (A.22) and(A.24) as follows.

tm ≥ tl′ (A.22′)

tm ≥ th′ (A.24′)

By the way, from wCP, given the opponent’s strategy x ∈ Δ({l′, h′}), thereexists a strictly dominant strategy x′ ∈ Δ({l′, h′}) against strategy k withl < k < h where x′ assigns probability x′

l′ to strategy l′ and x′h′ to h′ with

x′h′ =

(gh′h′ − gl′h′)

(gl′l′ − gh′l′) + (gh′h′ − gl′h′), x′

l′ = 1− x′h′.

25The proof is very similar to that of Proposition 7 but we cannot apply the same argumentbecause the game we consider here does not have a specific payoff function as in the minimum-effortgame. To compensate it, we use wCP and MRPD.

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tm

tl′

th′

tm

1

xh′10 x′h′

Figure 12: Expected payoffs given the opponent’s strategy x ∈ Δ({m,m})

Together with (A.22′) and (A.24′), for any m ≤ l′ or m ≥ h′, strategy x′ ∈Δ({l′, h′}) strictly dominates strategy k with l < k < h. To understandthis easily, see Figure 12 where the horizontal line is the probability assignedto strategy h′ and the vertical line is the parameterized payoffs given theopponent’s strategy m. Note that by the normalization, the payoffs withstrategy k is zero and the payoff differences gkm−gh′m, gkl′−gh′l′, gkm−gl′m, andgkh′ − gl′h′ are all one. Also note that due to wCP, the expected payoff of purestrategy l′ or h′ is higher than 0 at point x′

h′ . In Figure 12, strategy x′ gives ahigher (positive) payoff than that of strategy k (zero profit) for any opponent’sstrategy x ∈ Δ({m,m}) with m ∈ {l + 1, . . . , l′} and m ∈ {h′, . . . , h − 1}.Since x is expressed by some weighted average of x ∈ Δ({m,m}) for anym ∈ {l + 1, . . . , l′} and m ∈ {h′, . . . , h − 1}, strategy x′ strictly dominatesstrategy k given the opponent’s strategy x. Note that x is any mixed strategyexcept for the interior of the simplex.

By Lemma 5, we show Proposition 11 in the following.

Proof. Fix x ∈ Δ\intΔ. From Lemma 5, since any pure strategy k ∈ A\supp(x) isstrictly dominated by some mixed strategy in Δ(supp(x)), it follows that br(x) ⊆supp(x). For x ∈ intΔ, since supp(x) = A, br(x) ⊆ supp(x) must hold.

Thus, if we consider a symmetric supermodular coordination game under wCPand MRPD, CP holds.

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