chapter 8 strategy and game theory nicholson and snyder, copyright ©2008 by thomson south-western....

Chapter 8

Strategy and Game Theory

Nicholson and Snyder, Copyright ©2008 by Thomson South-Western. All rights reserved.

Basic Concepts

• In a strategic setting, a person may not always have an obvious choice of what is best– may depend on the actions of another person

Basic Concepts

• Two basic tasks when using game theory to analyze an economic situation– distill the situation into a simple game– solve the game

• results in a prediction about what will happen

Basic Concepts

• A game is an abstract model of a strategic situation

• Three elements to a game– players– strategies– payoffs

Players

• Each decision maker is a player– may be individuals, firms, countries, etc.– have the ability to choose from among a set of

possible actions

Strategies

• Each course of action open to a player is a strategy– it may be a simple action or a complex plan of

action

– Si is the set of strategies open to player i

– si is the strategy chosen by player i, si Si

Payoffs

• Payoffs are measured in levels of utility obtained by the players– players are assumed to prefer higher payoffs

to lower ones

– u1(s1,s2) denotes player 1’s payoff assuming she follows s1 and player 2 follows s2

• u2(s2, s1) would be player 2’s payoff under the same circumstances

Prisoners’ Dilemma

• The Prisoners’ Dilemma is one of the most famous games studied in game theory

• Two suspects are arrested for a crime

• The DA wants to extract a confession so he offers each a deal


• The Deal– “if you fink on your companion, but your

companion doesn’t fink on you, you get a one-year sentence and your companion gets a four-year sentence”

– “if you both fink on each other, you will each get a three-year sentence”

– “if neither finks, we will get tried for a lesser crime and each get a two-year sentence”


• There are 4 combinations of strategies and two payoffs for each combination– useful to use a game tree or a matrix to show

the payoffs• a game tree is called the extensive form• a matrix is called the normal form

Extensive Form for the Prisoners’ Dilemma

• Each node represents a decision point

• The dotted oval means that the nodes for player 2 are in the same information set– player 2 doesn’t know

player 1’s move

Normal Form for the Prisoners’ Dilemma

• Sometimes it is more convenient to represent games in a matrix

Nash Equilibrium

• Nash equilibrium involves strategic choices that, once made, provide no incentives for players to alter their behavior– best choice for each player given the other

players’ equilibrium strategies

Nash Equilibrium

• si is the best response for player i to rivals’ strategies s-i, denoted si BRi(s-i) if

ui(si,s-i) ui(s’i,s-i) for all s’i Si

• A Nash equilibrium is a strategy profile (s*1,s*2,…s*n) such that s*i is a best response to other players’ equilibrium strategies, s*-i or

s*i BRi(s*-i)

Nash Equilibrium

• In a 2-player game, (s*1,s*2) is a Nash equilibrium if

u1(s*1,s*2) u1(s1,s*2) for all s1 S1

u2(s*2,s*1) u2(s2,s*1) for all s2 S2

Nash Equilibrium

• Drawbacks to Nash equilibrium– there may be multiple Nash equilibria– it is unclear how a player can choose a best-

response strategy before knowing how rivals will play

Nash Equilibrium in Prisoners’ Dilemma

• Finking is player 1’s best response to player 2’s finking– the same logic applies for player 2

Nash Equilibrium in Prisoners’ Dilemma

• A quick way to find the Nash equilibria is to underline the best-response payoffs– the Nash equilibria correspond to the boxes in

which every player’s payoff is underlined

Dominant Strategies

• A strategy that is a best response to any strategy the other players might choose is called a dominant strategy– finking is a dominant strategy for both players

s*i BRi(s-i) for all s-i

• When a dominant strategy exists, it is the unique Nash equilibrium

Battle of the Sexes

• A wife and husband may either go to the ballet or to a boxing match– both prefer spending time together– the wife prefers ballet and the husband

prefers boxing

Battle of the Sexes

• There are two Nash equilibria– both going to the ballet– both going to boxing

• There is no dominant strategy

Rock, Paper, Scissors

• Two players simultaneously display one of three hand signals– rock breaks scissors– scissors cut paper– paper covers rock

Rock, Paper, Scissors

• None of the strategies is a Nash equilibrium

Mixed Strategies

• When a player chooses one action of another with certainty, he is following a pure strategy

• Players may also follow mixed strategies– randomly select from several possible actions

Mixed Strategies

• Reasons for studying mixed strategies– some games have no Nash equilibria in pure

strategies but will have one in mixed strategies

– strategies involving randomization are familiar and natural in certain settings

– it is possible to “purify” mixed strategies

Mixed Strategies

• Suppose that player i has a set of M possible actions, Ai = {a1

i,…ami,…,aM

i}

– a mixed strategy is a probability distribution over the M actions, si = (1

i,…,mi,…,M

i)

• 0 mi 1

1i+…+m

i+…+Mi = 1

Mixed Strategies

• A pure strategy is a special case of a mixed strategy– only one action is played with positive

probability

• Mixed strategies that involve two or more actions being played with positive probability are called strictly mixed strategies

Expected Payoffs in the Battle of the Sexes

• Suppose the wife chooses mixed strategy (1/9,8/9) and the husband chooses (4/5,1/5)

• The wife’s expected payoff is

boxing)(boxing,51

98

ballet)(boxing,54

98

boxing)(ballet,51

91

ballet)(ballet,54

91

11

11

uu

uu

4516

151

98

054

98

051

91

254

91


• Suppose the wife chooses mixed strategy (w,1-w) and the husband chooses (h,1-h)– the wife plays ballet with probability w and the

husband with probability h

• Her expected payoff becomes

boxing)(boxing,11ballet)(boxing,1

boxing)(ballet,1ballet)(ballet,

11

11

uhwuhwuhwuhw

11101012 hwhwhwhw

hwwh 31


• The wife’s best response depends on h– if h < 1/3, she should set w = 0– if h > 1/3, she should set w = 1– if h = 1/3, her expected payoff is the same no

matter what value of w she chooses


• The husband’s expected payoff is2 – 2h – 2w + 3hw

– when w < 2/3, he should set h = 0– when w > 2/3, he should set h = 1– when w = 2/3, his expected payoff is the same

no matter what value of h he chooses


w

m

1/3

1/3

2/3

2/3

1

1

Wife’s best response, (BR1)

Husband’s best response, (BR1)

There are three Nash equilibria

Mixed Strategies• A player randomizes over only those

actions among which he or she is indifferent

• One player’s indifference condition pins down the other player’s mixed strategy

Existence

• Nash proved the existence of a Nash equilibrium in all finite games– the existence theorem does not guarantee the

existence of a pure-strategy Nash equilibrium• it does guarantee that, if a pure-strategy Nash

equilibrium does not exist, a mixed-strategy Nash equilibrium does

Continuum of Actions

• Some settings are more realistically modeled via a continuous range of actions

• Using calculus to solve for Nash equilibria makes it possible to analyze how the equilibrium actions vary with changes in underlying parameters

Tragedy of the Commons

• The “Tragedy of the Commons” describes the overuse that arises when scarce resources are treated as common property– two herders decide how many sheep to graze

on the village commons– the commons is quite small and can rapidly

succumb to overgrazing


• Let qi = the number of sheep chosen by herder i

• Suppose that the per-sheep value of grazing on the commons is

v(q1,q2) = 120 – (q1 + q2)


• The normal form is a listing of the herders’ payoff functions

u1(q1,q2) = q1v(q1,q2) = q1(120 – q1 – q2)

u2(q1,q2) = q2v(q1,q2) = q2(120 – q1 – q2)

Tragedy of the Commons• To solve for the Nash equilibrium, we

solve herder 1’s maximization problem and get his best-response function

212

1 260 qBR

qq

• Similarly

121

2 260 qBR

qq

Tragedy of the Commons• The Nash equilibrium will satisfy both best-

response functions simultaneously

260

21

60 11

qq

q*1 = 40 q*2 = 40

Tragedy of the Commons• Suppose the per-sheep value of grazing

rises for herder 1– would result in more sheep for herder 1 and

fewer for herder 2

Tragedy of the Commons• The Nash equilibrium is not the best use of

the commons– if both herders grazed 30 sheep each, their

payoffs would rise

• Solving a joint-maxizimization problem will lead to the higher payoffs

Sequential Games

• In some games, the order of moves matters– a player that can move later in the game can

see how others have played up to that point

Sequential Battle of the Sexes

• Suppose the wife chooses first and the husband observes her choice before making his– her possible strategies haven’t changed– his possible strategies have expanded

• for each of his wife’s actions, he can choose one of two actions

Sequential Battle of the Sexes• The husband’s

decision nodes are not gathered together– he observes his

wife’s move– he knows which

decision node he is on before moving

Subgame-Perfect Equilibrium

• Subgame-perfect equilibrium rules out empty threats by requiring strategies to be rational even for contingencies that do not arise in equilibrium– a subgame is a part of the extensive form

beginning with a decision node and including everything to the right of it

– a proper subgame starts at a decision node not connected to another in an information set

Proper Subgames in the Battle of the Sexes

• The simultaneous game has only one proper subgame

Proper Subgames in the Battle of the Sexes

• The sequential games has three proper sub-games

Subgame-Perfect Equilibrium

• A subgame-perfect equilibrium is a strategy profile (s*1,s*2,…,s*n) that constitutes a Nash equilibrium for every proper subgame– a subgame-perfect equilibrium is always a

Nash equilibrium

• The sub-game perfect equilibrium rules out any empty threat in a sequential game

Backward Induction

• A shortcut for finding the perfect-subgame equilibrium directly is to use backward induction– working backwards from the end of the game

to the beginning

Backward Induction

• First, compute the Nash equilibria for the bottommost subgames at the husband’s decision nodes

• Next, substitute his equilibrium strategies for the subgames themselves

• The resulting game is a simple decision problem for the wife

Backward Induction

Repeated Games

• In many real-world settings, players play the same game over and over again– the simple constituent game that is played

repeatedly is called the stage game

• Repeated play opens up the possibility of cooperation in equilibrium– players can adopt trigger strategies

• cooperate as long as everyone else does

Finitely Repeated Games

• Repeating a stage game for a known, finite number of times may not increase the possibility for cooperation

• Selten’s theorem– if the stage game has a unique Nash

equilibrium, then the unique subgame-perfect equilibrium of the finitely repeated game is to play the Nash equilibrium every period


• If the stage game has multiple Nash equilibria, it may be possible to achieve cooperation in a finitely repeated game– players can use trigger strategies to maintain

cooperation• threaten to play the Nash equilibrium that yields a

worse outcome for the player who deviates from cooperation


• For cooperation to be sustained in a subgame-perfect equilibrium, the stage game must be repeated enough that the punishment for deviation is severe enough to deter deviation– the more repetitions, the more severe the

possible punishment, and the greater the level of cooperation


• Folk theorem for finitely repeated games– Suppose that the stage game has multiple Nash

equilibria and no player earns a constant payoff across all equilibria

– Any feasible payoff in the stage game greater than the player’s pure-strategy minmax value can be approached arbitrarily closely by the player’s per-period average payoff in some subgame-perfect equilibrium of the finitely repeated game for large enough T


• A feasible payoff is one that can be achieved by some mixed-strategy profile in the stage game

• The minmax value is the lowest payoff a player can be held to if all other players work against him but he is able to choose a best response to them

Infinitely Repeated Games

• A folk theorem will apply to infinitely repeated games even if the underlying stage game has only one Nash equilibrium

• To circumvent the problem with adding up payoffs across periods, we will use a discount factor ()– let measure the value of a payoff unit received

one period in the future rather than today


• Players can sustain cooperation in infinitely repeated games by using trigger strategies– the trigger strategy must be severe enough to

deter deviation


• Suppose both players follow a trigger strategy in the Prisoners’ Dilemma– if both players are silent every period, the payoff

over time would be

Veq = 2 + 2 + 22 + … = 2/(1–)– if a player deviates and then the other finks

every period, that player’s payoff is

Vdev = 3 + 1 + 12 + … = 3 + /(1–)


• The trigger strategies form a perfect-subgame equilibrium if Veq Vdev

2/(1–) 3 + /(1–)

1/2


• The trigger strategy in which players revert to the harshest punishment possible is called the grim strategy– revert to the stage-game Nash equilibrium

forever– elicits cooperation for the lowest value of for

any strategy

• A tit-for-tat strategy involves only one round of punishment for cheating


• As approaches 1, grim-strategy punishments become infinitely harsh– involve an unending stream of undiscounted

losses


• The folk theorem for infinitely repeated games– Any feasible payoff in the stage game greater

than the player’s minmax value can be obtained as the player’s normalized payoff (normalized by multiplying by 1-) in some subgame-perfect equilibrium of the infinitely repeated game for close enough to 1.


• Differences with the folk theorem for finitely repeated games– the limit involves increases in rather than in

the number of periods, T– the folk theorem for infinitely repeated games

holds even if there is only one Nash equilibrium in the stage game

Incomplete Information

• Matters become more complicated if some players have information about the game that others do not

• Players that lack full information will try to use what they do know to make inferences about what they do not

Simultaneous Bayesian Games

• Consider a simultaneous-move game where player 1 has private information but player 2 does not

• We can model private information by introducing player types

Player Types and Beliefs

• Player 1 can be of a number of possible types, t

• Player 2 is uncertain about t– must decide strategy based on her beliefs

about t


• The game begins at a chance node, at which a value of tk is drawn for player 1 from a set of possible types, T = {t1,…,tk,…,tK}

– Pr(tk) = probability of drawing type tk

– player 1 sees which type is drawn– player 2 does not see which type is drawn

• only knows the probabilities


• Since player 1 observes t before moving, his strategy can be conditioned on t– let s1(t) be 1’s strategy contingent on his type

– player 2’s strategy is an unconditional one, s2


• Player 1’s type may affect player 2’s payoff in two ways– directly– indirectly through player 1’s strategy


• All payoffs are known except for 1’s payoff when 1 chooses U and 2 chooses L– depends on his type, t

• t can be 6 or 0, with equal probability

Bayesian-Nash Equilibrium

• Equilibrium requires that – 1’s strategy be the best response for each

and every one of his types– 2’s strategy maximize an expected payoff

• the expectation is taken with respect to her beliefs about 1’s type

Bayesian-Nash Equilibrium

• In a two-player simultaneous move game in which player 1 has private information, a Bayesian-Nash equilibrium is a strategy profile (s*1(t),s*2) such that

u1(s*1(t),s*2,t) u1(s’1,s*2,t) for all s’1 S1

and 1

'2

*1

'22

*1

*22 all for ,,Pr,,Pr Ssttssutttssut

Ttkkk

Ttkkk

kk

Bayesian-Nash Equilibrium• Two possible

candidates for an equilibrium in pure strategies

1 plays (U|t=6, D|t=0) and 2 plays L

1 plays (D|t=6, D|t=0) and 2 plays R

Bayesian-Nash Equilibrium• This is not an

equilibrium:1 plays (U|t=6, D|t=0)

and 2 plays L

• This is a Bayesian-Nash equilibrium:

1 plays (D|t=6, D|t=0) and 2 plays R

Signaling Games

• A signaling game occurs in a sequential game with private information– the informed player (player 1 ) takes an action

that is observable to player 2 before 2 moves• player 1’s action can serve as a signal to player 2

that can be used to update her beliefs about player 1’s type

Job-Market Signaling

• Player 1 can be of two types– highly skilled (t = H)– low skilled (t = L)

• Player 2 is a firm considering hiring player 1– low-skilled worker generates no revenue for

the firm– high-skilled worker generates revenue of


• Regardless of type, if hired the worker gets paid a wage of w

• The firm cannot observe the worker’s skill, only his education level– let cL be the cost of obtaining an education for

the low type

– let cH be the cost of obtaining an education for the high type

• Assume > w > 0 and cL < cH


• Player 1 observes his type before moving

• Player 2 only observes player 1’s action (education signal) before moving

• Player 2’s expected payoff from playing J is

Pr(H|E)( - w) + Pr(L|E)(-w) = Pr(H|E) - w


• Player 2’s best response is J if and only if Pr(H|E) w/

Bayes’ Rule

LLEHHE

HHEEH

PrPrPrPr

PrPrPr

LLNEHHNE

HHNENEH

PrPrPrPr

PrPrPr

Bayes’ Rule

• When player 1 plays a pure strategy, Bayes’ rule often gives a simple result

• Suppose that Pr(E|H) = 1 and Pr(E|L)=0

1

Pr0Pr1Pr1

Pr

LHH

EH

Bayes’ Rule

• Suppose that Pr(E|H) = 1 and Pr(E|L)=1

HLH

HEH Pr

Pr1Pr1Pr1

Pr

Perfect Bayesian Equilibrium

• A perfect Bayesian equilibrium consists of a strategy profile and a set of beliefs such that at each information set– the strategy of a player moving there

maximizes his expected payoff• the expectation is taken with respect to his beliefs

– the beliefs of the player moving there are formed using Bayes’ rule


• Signaling games may have multiple equilibria– they can be of three types

• separating• pooling• hybrid


• In a separating equilibrium, each type of player chooses a different action– player 2 learns player 1’s type with certainty

after 1 moves


• In a pooling equilibrium, different types of player 1 choose the same action– observing player 1’s move provides player 2

with no additional information


• In a hybrid equilibrium, one type of player 1 plays a strictly mixed strategy– player 2 learns a little about player 1’s type

but doesn’t learn it with certainty– player 2 may respond by also playing a mixed

strategy

Cheap Talk

• Education must be more costly for the low-skilled worker or else skill levels could not be separated in equilibrium

• In the real world, most information is communicated at low or no cost (“cheap talk”)

Cheap Talk

• Suppose that player 1’s strategy space consists of messages sent costlessly to player 2– player 1 communicates to 2– player 2 takes some action affecting both

players’ payoffs

Cheap Talk

• The maximum amount of information that can be contained in player 1’s message will depend on how well-aligned the players’ payoff functions are– as preferences diverge, messages become

less and less informative

Experimental Games

• Experimental economics explores how well economic theory matches the behavior or experimental subjects in a laboratory setting– play in the Prisoners’ Dilemma converged to

the Nash equilibrium as subjects gained experience

– results from the Ultimatum game suggests that fairness matters

Important Points to Note:

• All games have the same basic components– players– strategies– payoffs– an information structure


• Games can be written down in normal form or extensive form

• Strategies can be simple actions, more complicated plans contingent on others’ actions, or even probability distributions over simple actions (mixed strategies)


• A Nash equilibrium is a set of strategies that are mutual best responses– a player’s strategy in a Nash equilibrium is

optimal given that all others play their equilibrium strategies

• A Nash equilibrium always exists in finite games– in mixed if not pure strategies


• Subgame-perfect equilibrium is a refinement of Nash equilibrium that helps to rule out equilibria in sequential games involving noncredible threats


• Repeating a stage game a large number of times introduces the possibility of using punishment strategies to attain higher payoffs than if the game is played once– if a finite game is repeated enough or if

players are sufficiently patient in an infinite game, a folk theorem holds implying that essentially any payoffs are possible


• In games of private information, one player knows more about his “type” than the other– players maximize their expected payoffs given

knowledge of their own types and beliefs about the others’


• In a perfect Bayesian equilibrium of a signaling game, the second mover uses Bayes’ rule to update his beliefs about the first mover’s type after observing the first mover’s action


• The frontier of game-theory research combines theory with experiments to determine – whether players who may not be hyperrational

come to play a Nash equilibrium– which particular equilibrium (if there are more

than one)– what path leads to the equilibrium

chapter 8 strategy and game theory nicholson and snyder, copyright ©2008 by thomson south-western....

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s i s i

player i s i

u i s i

denoted s i br i s

nash equilibrium s i

strategy profile s

complex plan of action

player game