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
Prisoners’ Dilemma
• 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”
Prisoners’ Dilemma
• 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
Expected Payoffs in the Battle of the Sexes
• 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
Expected Payoffs in the Battle of the Sexes
• 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
Expected Payoffs in the Battle of the Sexes
• 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
Expected Payoffs in the Battle of the Sexes
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
Tragedy of the Commons
• 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)
Tragedy of the Commons
• 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
• Similarly
121
2 260 qBR
Tragedy of the Commons• The Nash equilibrium will satisfy both best-
response functions simultaneously
260
21
60 11
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
Sequential Battle of the Sexes
• There are three pure-strategy Nash equilibria– wife plays ballet, husband plays (ballet | ballet,
ballet | boxing)– wife plays ballet, husband plays (ballet | ballet,
boxing | boxing)– wife plays boxing, husband plays (boxing | ballet,
boxing | boxing)
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
Finitely Repeated Games
• 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
Finitely Repeated Games
• 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
Finitely Repeated Games
• 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
Finitely Repeated Games
• 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
Infinitely Repeated Games
• Players can sustain cooperation in infinitely repeated games by using trigger strategies– the trigger strategy must be severe enough to
deter deviation
Infinitely Repeated Games
• 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–)
Infinitely Repeated Games
• The trigger strategies form a perfect-subgame equilibrium if Veq Vdev
2/(1–) 3 + /(1–)
1/2
Infinitely Repeated Games
• 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
Infinitely Repeated Games
• As approaches 1, grim-strategy punishments become infinitely harsh– involve an unending stream of undiscounted
losses
Infinitely Repeated Games
• 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.
Infinitely Repeated Games
• 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
Player Types and Beliefs
• 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
Player Types and Beliefs
• 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 Types and Beliefs
• Player 1’s type may affect player 2’s payoff in two ways– directly– indirectly through player 1’s strategy
Player Types and Beliefs
• 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
Job-Market Signaling
• 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
Job-Market Signaling
• 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
Job-Market Signaling
• 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
Perfect Bayesian Equilibrium
• Signaling games may have multiple equilibria– they can be of three types
• separating• pooling• hybrid
Perfect Bayesian Equilibrium
• In a separating equilibrium, each type of player chooses a different action– player 2 learns player 1’s type with certainty
after 1 moves
Perfect Bayesian Equilibrium
• 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
Perfect Bayesian Equilibrium
• 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
Important Points to Note:
• 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)
Important Points to Note:
• 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
Important Points to Note:
• Subgame-perfect equilibrium is a refinement of Nash equilibrium that helps to rule out equilibria in sequential games involving noncredible threats
Important Points to Note:
• 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
Important Points to Note:
• 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’
Important Points to Note:
• 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
Important Points to Note:
• 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
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