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### Game Theory

6/3/10

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, siSi

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

- useful to use a game tree or a matrix to show the payoffs

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 siBRi(s-i) if
ui(si,s-i) ui(s’i,s-i) for all s’iSi

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

- finking is a dominant strategy for both players
- 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 = {a1i,…ami,…,aMi}
- a mixed strategy is a probability distribution over the M actions, si = (1i,…,mi,…,Mi)
- 0 mi 1
- 1i+…+mi+…+Mi = 1

- a mixed strategy is a probability distribution over the M actions, si = (1i,…,mi,…,Mi)

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

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

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 is
2 – 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

There are three Nash equilibria

Expected Payoffs in the Battle of the Sexesm

Husband’s best response, (BR1)

1

2/3

1/3

Wife’s best response, (BR1)

w

2/3

1

1/3

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

- the existence theorem does not guarantee the existence of a pure-strategy Nash equilibrium

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

- Similarly

Tragedy of the Commons

- The Nash equilibrium will satisfy both best-response functions simultaneously

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

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

- players can adopt trigger strategies

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

- players can use trigger strategies to maintain 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–)

- if both players are silent every period, the payoff over time would be

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

- depends on his type, t

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

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

- the informed player (player 1 ) takes an action that is observable to player 2 before 2 moves

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

- 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

Bayes’ Rule

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

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

- the strategy of a player moving there maximizes his expected payoff

Perfect Bayesian Equilibrium

- Signaling games may have multiple equilibria
- they can be of three types
- separating
- pooling
- hybrid

- they can be of three types

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

Problems

- Consider the following scenario. Chris and Pat are coworkers who race each other from the office to a nearby bar every day after work. Each must choose independently whether to take the highway (H) or local streets (L). The highway is typically quicker, but sometimes is congested. Neither knows what the other has chosen before make his/ her own choice. If both take the highway, they arrive quickly and celebrate their luck by having 7 drinks each. If both take local roads then they arrive later, but they still arrive together and each has 5 drinks. If one takes the highway and the other takes the local roads, the person who takes the highway arrives first and has a drink while waiting for the one who took the local roads. When the second person arrives, he/ she is annoyed that the first person started to drink before they arrived and each only has one additional drink before heading home. Both have the same utility function U(d)=d, where d is the total number of drinks consumed.

Problems

- Represent the game in both normal (payoff matrix) and extensive (game tree) form.
- Does either player have a dominant strategy?
- Find the complete set of Nash Equilibria (pure and mixed strategy) for this game. What is expected payoff to each player in each equilibrium?

Problems

- Consider the following game. Find the compete set of Nash Equilibria (in pure and mixed strategies). (4 possiblepoints)
Player 2

L R

U 6,6 12,3

Player 1

D 3,12 9,9

- For what values of δ is playing (D,R) forever a SPNE of the repeated game?

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