static games of complete information

GAMESEQUILIBRIUM CONCEPTS

THE TRAGEDY OF THE COMMONSMULTIPLE NASH EQUILIBRIA

NASH THEOREM

3. Lectures onSTATIC GAMES OF COMPLETE

INFORMATION

Luigi Buzzacchie-mail: [email protected]

Politecnico di TorinoDIST

A.A. 2013/14

v. 30 03 14 Static games of complete information

mailto:[email protected]



NASH THEOREM

Outline of the chapter

1 GAMESNormal-form representation of games

2 EQUILIBRIUM CONCEPTSElimination of strictly dominated strategiesIterated elimination of strictly dominated strategiesNash equilibrium

3 THE TRAGEDY OF THE COMMONS

4 MULTIPLE NASH EQUILIBRIAMixed strategies

5 NASH THEOREM




NASH THEOREM

Normal-form representation of games

Definitions /1

Game theory is a set of tools that is used to model the behaviour orchoices of players (consumers, firms, etc.) when the payoff (profit) of achoice depends on the choice of other players. Recognized payoffinterdependency gives rise to interdependent decision making orstrategic interactionThe optimal choice of a player will depend on her expectation of thechoices of others playing the same game. How expectations of everyplayer are formed?




NASH THEOREM


Definitions /2

Our game-theoretic analysis builts on three fundamental assumptions:1 Rationality. We assume that a player’s preferences can be represented

by a utility function or payoff function. Game theory assumes thatplayers are interested in maximizing their payoffs. A utility functionsimply assigns an index number to each outcome with the property thathigher index numbers are assigned to outcomes that are more preferred.In game theory it is common to refer to a player’s utility function as herpayoff function. Payoffs for firms are simply profits or expected profits

2 Common information. Every player knows i) the structure of thegame and ii) that his opponents are rational. This information iscommon knowledge. An information is commonly known when all ofthe players know that information; moreover, they know that theiropponents know that they know that information; moreover, they knowthat their opponents know that they know that their opponents know. . . , and so on

3 Non-cooperation. Players are unable to sign credible contracts beforethe beginning of the game




NASH THEOREM


Normal-form representation of games /1

The normal form specification of a game specifies:1 the number (n) of the players in the game2 the set of (pure) strategies (Si) available to each player; a strategy is

identified by si ∈ Si (i = 1 . . . n). Actions and strategies coincide only instatic games of complete information

3 a payoff function (ui = ui(s)) for each player. Such function assigns eachplayer i a level of utility for each profile/combination of strategies s,s ∈ S1 × S2 × . . .× Sn

The normal form representation of a game is consequently:G = {S1 . . . Sn; u1 . . . un}




NASH THEOREM


Normal-form representation of games /2

In a static game players choose their strategies simultaneously.Simultaneously means that none of the players can benefit of theinformation concerning the choice of the rivals (logical vs.chronological simultaneity)

An equilibrium concept is a function that associates to a game aspecific combination of strategiesΓ : G→ {s1 . . . sn}The combination selected by the function Γ is the one which iscomposed by strategies that make the correspondent player satisfied

⇒What is satisfaction?




NASH THEOREM

Elimination of strictly dominated strategiesIterated elimination of strictly dominated strategiesNash equilibrium

The prisoner’s dilemma /1

Player 2Mum Fink

Player 1 Mum −1,−1 −9, 0

Fink 0,−9 −6,−6

Dominated strategy:s′i is dominated if ∃ s′′i | ui(s′′i , s−i) > ui(s′i , s−i) ∀ s−i

Strict and weak dominance




NASH THEOREM


The prisoner’s dilemma /2

Mum is dominated by Fink: the combination of strategies {Fink; Fink}solves the gameSolving a game means providing a forecast of the outcome (of theequilibrium)In general, we have a prisoner’s dilemma game when:D(efection) > C(ooperation) > P(unishment) > S(ucker’s payoff)

Player 2Mum Fink

Player 1 Mum C,C S,D

Fink D, S P,P

The free-riding problem can be interpreted as a prisoner’s dilemma




NASH THEOREM


Iterated elimination of strictly dominated strategies /1

In this game, several combinations of strategies are all composed bynon-dominated strategies

Player 2Left Center Right

Player 1 Up 1, 0 1, 2 0, 1

Down 0, 3 0, 1 2, 0

Only Up, Center, however, survives to an iterative process ofelimination of dominated strategies




NASH THEOREM


Iterated elimination of strictly dominated strategies /2

Is that procedure a good representation of a rational choice? A rationalplayer never plays strictly dominated strategies. This equilibriumconcept need adequate assumptions concerning the rationality of theopponent (common knowledge)Despite the fact that iterated elimination of dominated strategies is a‘less demanding’ concept, the largest part of conceivable games stillcannot be ‘solved’


Player 1Left 0, 4 9, 0 5, 3

Center 4, 0 4, 4 5, 3

Right 3, 5 2, 5 6, 6




NASH THEOREM


Best responses and Nash equilibrium

A reaction function is a function defined in the strategy space as:Ri(s−i) = argmaxsi [ui(si, s−i)]

A best response si is a strategy which is a specific value of the reactionfunction, i.e.si = Ri(s−i)

A strategically stable (self-enforcing) outcome is obtained when noplayer has anything to gain by changing his own strategy unilaterally

A strategy combination is a Nash equilibrium if its outcome isstrategically stable




NASH THEOREM


Nash equilibrium

In the game G, the strategy combination sN = {sN1 . . . sN

n } is said to be aNash equilibrium if:

ui(sNi , sN−i) ≥ ui(si, sN

−i) ∀si, ∀iThis happens when sN

i solves the problem:maxsi [ui(si, sN

−i)]


Player 1Left 0, 4 9, 0 5, 3

Center 4, 0 4, 4 5, 3

Right 3, 5 2, 5 6, 6

Notice that every strategy is a best response to some strategy of therival. This is a sufficient (not necessary) condition for not having anydominated strategy (see below)




NASH THEOREM


Equilibrium concepts hierarchy

Proposition A. In the normal form game G, if iterated elimination ofstrictly dominated strategies eliminates all but one strategy for everyplayer, then the survived strategy combination is the unique Nashequilibrium of the game.Proposition B. In the normal form game G, every strategy combinationwhich is a Nash equilibrium survives iterated elimination of strictlydominated strategies

An example of a Nash equilibrium that does not survive an iteratedelimination of weakly dominated strategies is the Prisoner’s Dilemmawhere the {Mum; Mum} payoff is 0 for both players




NASH THEOREM

The tragedy of the commons [Hardin, 1968] /1

It is a problem of public goods exploitationn farmers own gi (i = 1 . . . n) animals which are grazed on the villagegreenThe cost per head of livestock is c

The revenue per head of livestock is:v(∑

i gi) v′ < 0 [and v′′ < 0]The strategic variable of the players is the number gi of animals they(simultaneously) decide to breedThe individual farmer’s payoff is consequently:ui(gi, g−i) = gi v(gi + g−i)− cgi




NASH THEOREM

The tragedy of the commons [Hardin, 1968] /2

The Nash equilibrium is defined by:g∗i = argmaxgi [giv(gi + g∗−i)− cgi] ∀iFOCs are:v(gi + g∗−i) + giv′(gi + g∗−i)− c = 0or, equivalently

v(G∗) +G∗v′(G∗)

n− c = 0

The social optimum needsG◦

= argmaxG[Gv(G)− cG]whose FOC isv(G

◦) + G

◦v′(G

◦)− c = 0

It can be easily obtained that G∗ > G◦

This effect exemplifies the so called free riding behaviour, whichemerges in presence of negative externalities (as is the exploitation ofpublic goods)




NASH THEOREM

Mixed strategies

Battle of the sexes and matching pennies

PatOpera Fight

Chris Opera 2, 1 0, 0

Fight 0, 0 1, 2

Battle of the sexes

EqualHeads Tails

Different Heads −1, 1 1,−1

Tails 1,−1 −1, 1

Matching pennies




NASH THEOREM

Mixed strategies

No unique equilibria and randomized actions

In both games the payoffs depend on the coordination of the actions ofthe players. As they disagree about the preferred outcome, this type ofgames are not characterized by unique equilibriaThe behavior of the rival is consequently uncertainWe extend the analysis by admitting the possibility that the playersrandomize their actionsA mixed strategy is a probability distribution:

{pi1 . . . piK} (

K∑k=1

pik = 1),

over the strategies in:Si = {si1 . . . siK}




NASH THEOREM

Mixed strategies

Mixed strategies /1

The basic idea is that the concept of dominance has to be redefinedonce we admit mixed strategiesWhen si is dominated, then no one (pure) strategy of the opponents s−i

(and consequently no one belief) can make si a best responseIt can be easily demonstrated that if si is dominated, then not even amixed strategy of the opponents can make si a best responseThe opposite is false if we consider only pure strategies . . .. . . but the opposite becomes true if we admit mixed strategiesIn particular, . . . if there is no belief that player i could hold about s−i

such that it would be optimal to play si, then there necessarily existsa strategy s′i (possibly mixed) that strictly dominates si




NASH THEOREM

Mixed strategies

Mixed strategies /2

In the following game, B is never a best response for every purestrategy played by 2, but neither T nor M dominate B. However, amixed strategy with pT = 1

2 and pM = 12 , strictly dominates the pure

strategy B (in expected value)

Player 2L R

Player 1T 3, · 0, ·M 0, · 3, ·B 1, · 1, ·

The Nash equilibrium concept can be obviously extended when mixedstrategies are allowedA pure strategy is only a particular mixed strategy




NASH THEOREM

Mixed strategies

Best responses to mixed strategies

In a two players game with two symmetric actions {a, b}, if player 1expects form player 2 a strategy {q, 1− q}, her payoff becomes:

E(u(a)) = qu(a, a) + (1− q)u(a, b)andE(u(b)) = qu(b, a) + (1− q)u(b, b)

Depending on q (the probability of a), the best response for player 1could be a or b

In particular, the threshold q′,

q′ =u(b, b)− u(a, b)

u(b, b) + u(a, a)− u(a, b)− u(b, a)

determines when the best response is a or the opposite (q ≶ q′). Whenq = q′, player 1 is indifferent between a and b




NASH THEOREM

Mixed strategies

Reaction functions for mixed strategies

In the case of a matching pennies game, for example, q′ = 12

In a similar way player 2 can calculate her threshold r′




NASH THEOREM

Mixed strategies

Mixed strategies /3

More in general, in a two-player game, if p1 and p2 are the probabilitydistributions over the two set of actions of the players composed of Jand K elements, then:

v1(p1, p2) = E(u1(p1, p2)) =

J∑j=1

K∑k=1

p1jp2ku1(s1j, s2k)

v2(p1, p2) = E(u2(p1, p2)) =

J∑j=1

K∑k=1

p1jp2ku2(s1j, s2k)

. . . and the definition of Nash equilibrium becomes:{v1(p∗1 , p∗2) ≥ v1(p1, p∗2)

v2(p∗1 , p∗2) ≥ v2(p∗1 , p2)∀ p1, p2




NASH THEOREM

Mixed strategies

Properties of mixed strategies equilibria

The condition for a mixed strategy p∗1 to be the best response to themixed strategy p∗2 , is that (from the FOC) p∗1j > 0 iif the pure strategy jis also best response to p2, or equivalently:

K∑k=1

p∗2ku1(s1j, s2k) ≥K∑

k=1

p∗2ku1(s1j′ , s2k) ∀j′

The interpretation of such property is the following. Expand v1 as:

v1(p1, p2) = p11

K∑k=1

p2ku1(s11, s2k) + . . . + p1J

K∑k=1

p2ku1(s1J, s2k)

The weighted sum above obviously increases when we increase theprobability for pure strategies j that present a higher payoff

K∑k=1

p2ku1(s1j, s2k) (at the same time decreasing the probabilities of

lower payoff strategies). The optimum is reached when only thestrategies with equal maximum expected payoff are randomized withpositive probability




NASH THEOREM

Mixed strategies

Fundamental Lemma

At the equilibrium, each player is indifferent between the equilibriummixed strategy and every pure strategy played with positive probability




NASH THEOREM

Mixed strategies

Example: controlling fare evasion

The game below illustrates the strategic interaction between a publictransport traveller (T) and the operator (O). P is the price of the ticket,F is the fine and c is the cost of controlling, with F > T > c

OControl No control

T Buy ticket −T,T − c −T,T p

Evade fare −F,F − c 0, 0 1− pq 1− q

The expected payoff for T is −pqT − p(1− q)T − (1− p)qFThe expected payoff for O is pq(T − c) + p(1− q)T + (1− p)q(F − c)

The Nash equilibrium is characterised by q∗ =TF

and p∗ =F − c

FAs F increases p∗ also increases, while q∗ decreasesAs T increases q∗ also increasesAs c increases p∗ decreases




NASH THEOREM

Possible equilibria in a 2× 2 static game

P2L R

P1 U x, · y, ·D z, · w, ·

x > z; y > w→ dominancex < z; y < w→ dominancex > z; y < wx < z; y > w

If {r, 1− r} and {q, 1− q} are the generic strategies of P1 and P2respectively; in the first and in the second case r∗ = r(q) is a constant(0 or 1); in the third and in the fourth cases the threshold q′ = w−y

x−z+w−yis the value where r∗ = 0 becomes r∗ = 1In the equality cases (x = z and/or y = w), one of the strategies weaklydominates the other one, so that the reaction function is L-shaped(q′ = 0 or q′ = 1)




NASH THEOREM

Nash theorem [1950]

Every couple of reaction function of the type illustrated above certainlyintersect in the r/q space within the square 0/1In the sixteen possible cases, we can then obtain:

1 a unique NE in pure strategies2 a unique NE in mixed strategies3 two pure-strategy NE and one NE in mixed strategies

The Nash theorem generalizes such result when the number of playersand the number of strategy combinations are finite:

. . . if n and Si are finite ∀i in G(S1 . . . Sn; u1 . . . un), then at least oneNE exists, possibly involving mixed strategies

Demonstration is based on the fixed point theorem:. . . if f (x) is continuous in [0, 1] ∃ x∗ → f (x∗) = x∗


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