week 1: game theory - university of warwick · game theory books for more detail on game theory i...

Reading Introduction PD More Examples Best Responses Summary

Week 1:Game Theory

Dr Daniel Sgroi

Reading: Osborne chapter 2;Snyder/Nicholson chapter 8, pp. 236–245.

With thanks to Peter J. Hammond.

EC202, University of Warwick, Term 2 1 of 29


Reading

Term 2’s lectures will start with Game Theory and will take usthrough most of the term. We will follow with lectures onuncertainty and information.The list of possible textbooks you could use is enormous. If youhave a text you like, keep using it! This course should cover mostof what you need especially in combination with the chapters onGame Theory and Asymmetric Information in most modernmicroeconomics books. Examples:

W. Snyder C.M. Nicholson: Intermediate Microeconomics and itsApplicationsA. Schotter: MicroeconomicsR.S. Pindyck D.L. Rubinfeld: Microeconomics (Chapter 13)D. Kreps: A Course on Microeconomic Theory (Chapters 11-15, abit too advanced)



Game Theory Books

For more detail on Game Theory I will normally refer to:

M. Osborne: An introduction to Game Theory. This is the closestto a course text for the Game Theory material.

Other game theory texts include:

K. Binmore: Fun and Games (a bit advanced)P. Dutta: Games and StrategiesR. Gibbons: A Primer in Game TheoryD. Kreps: Game Theory and Economic Modelling (non-technicalessay on the current state of game theory)E. Ramusan: Games and InformationJ. Watson: Game TheoryJ. Tirole: The Theory of Industrial Organization (Appendix onGame Theory - concise)



Strategic Settings

A strategic setting ariseswhen one decision maker’s choice of what seems bestdepends on the actions of one or more other persons.

For example, each shepherd i ’s optimal choice s∗idepends on the total number

∑j 6=i sj of sheep

that all other shepherds graze on the common.

In a static game, the players independently makea single once-and-for-all choice of action.

Given the actions [“strategies”] that all players choose,some outcome [“consequence”] will occurthat affects each player’s well-being.



Three Key Ingredients

There are three key ingredients in the definition of a game:

1. a set of players;

2. a strategy set from which each player can choose;

3. a payoff function specifying what each player “gets paid”.



Ingredient 1: The Player Set

There is fixed finite set N of n players,often taken as N = {1, 2, . . . , n}.Each decision maker is a player.They may be individuals, households, firms,national or regional governments, etc.

When n = 1 there is a one-person game,which is the subject of single-person decision theory.Typically economists assume that single agents like thismaximize the expected value of a utility function.

Game theory amounts to multi-person decision theory.



Ingredient 2: Strategy Profiles

A player’s strategy in the gameis a complete course of action open to that player.

It may be simple, like choosing to drive on the left or the right.

It may be a real number, like a firm’s output.

It may be tremendously complex,like a computer programme that plays chess.

Each player i in the player set Nis assumed to have their own strategy set Si .The players i ∈ N simultaneously and independently choosetheir actions or strategies si from their respective strategy sets Si .In particular, each player i chooses a strategy si ∈ Siwithout observing the other players’ choices sj (j ∈ N \ {i}).Together, the choices of all the players i in set N determinea strategy profile sN = 〈si 〉i∈N = (s1, s2, . . . , sn).



Ingredient 3: Payoffs

The players’ chosen strategy profile sN resultsin an outcome or consequence yi (s

N) for each player i ∈ N.For example, profits accruing to each shepherd.

We assume that each player i ∈ N has preferencesover random consequences which are represented bythe expected value of some utility function vi (yi ).

Game theorists usually replace each player i ’sindirect function vi (yi (s

N)) of a functionby the equivalent direct payoff function ui (s

N)defined on the space SN of strategy profiles sN .



Summary

To summarize, a game will be a triple 〈N,SN , uN〉 where:

1. N = {1, 2, . . . , n} is the set of players;

2. SN =∏n

i=1 Si is the set of all possibleplayers’ strategy profiles sN = (s1, s2, . . . , sn) = 〈si 〉i∈N ;

3. uN = (u1, u2, . . . , un) = 〈ui 〉i∈N is the profileof all the players’ different payoff functions sN 7→ ui (s

N).

Sometime this is called a game in normal or strategic form.This contrasts with extensive form games we shall consider later.



Prisoner’s Dilemma: Players and Strategies

Prisoner’s Dilemma is probably the most well known examplein game theory, with some very general applications.Suppose two individuals are suspected of a crime.They are questioned in separate rooms at the police station.Each faces a choice between two actions:

1. to say nothing to the investigators, also called “mum” (M);

2. or to rat on the other suspect, also called “fink” (F ).



PD: Outcomes

Each suspect’s payoff is determined as follows:

• if both stay mum, then both get 1 year in prisonsince the evidence only supports a minor offence;

• if, say, individual 1 stays mum while individual 2 finks,then individual 1 gets 9 years in prisonwhile individual 2 walks away free(vice versa if 1 finks while individual 2 stays mum);

• if both fink, then both get only 6 years.

Assume each year in prison is worth −1 unit of payoff.



PD: Normal Form

The normal form game representation of the Prisoner’s Dilemma is:

players: N = {1, 2}strategy sets: Si = {M,F} for i ∈ {1, 2};

payoffs: Let ui (s1, s2) be the payoff to player iif player 1 chooses s1 and player 2 chooses s2.

We can then write payoffs as:

u1(M,M) = u2(M,M) = −1; u1(F ,F ) = u2(F ,F ) = −6

u1(M,F ) = u2(F ,M) = −9; u1(F ,M) = u2(M,F ) = 0



Bimatrix GamesDefinition: A finite game has a finite number of players,with finite strategy sets Si for all players i ∈ N.Any two-person finite game has a convenient representationas a (bi-)matrix that captures all the game’s relevant information:

Rows: Each row represents a different strategyfor the row player 1 (called Rowe, or Rowena?).The number of rows in the bimatrix equals #S1,the number of elements in player 1’s strategy set.

Columns: Each column represents a different strategyfor the column player 2 (called Colm, or Colin?).So the bimatrix has #S2 columns.

Bimatrix entries: The entry in row i and column j of this bimatrixis the payoff pair (u1(i , j), u2(i , j)),where uk(i , j) is player k ’s payoff (k = 1, 2)when player 1 chooses row iand player 2 chooses column j .



PD in Bimatrix Form

Each player has two actions, M (mum) and F (fink).So the matrix has two rows (for player 1)and two columns (for player 2).Using the payoffs specified previously,the bimatrix representation is:

Player 2

M F

M −1 −9−1 0

Player 1 F 0 −6−9 −6

All the relevant information appears in this bimatrix.



BoS

This game was originally called “battle of the sexes”,involving a man who would like to watch boxing,and a woman who would prefer ballet.Martin Osborne prefers Bach or Stravinsky.Here it could be an evening out in Birmingham, or one in Stratford.

players: N = {1, 2}strategy sets: Si = {B, S} for i ∈ {1, 2};

payoffs: Let ui (s1, s2) be the payoff to player iif player 1 chooses s1 and player 2 chooses s2.

We can then write payoffs as:

u1(B,B) = u2(S , S) = 2; u2(B,B) = u1(S , S) = 1

u1(B, S) = u2(B,S) = 0; u1(S ,B) = u2(S ,B) = 0



BoS in Bimatrix Form

Player 2

B S

B 2 01 0

Player 1 S 0 10 2



Matching Pennies

Here the two players 1 and 2 can each choosewhich face of a coin to show the other— either heads (H) or tails (T ).The payoff bimatrix is

Player 2

H T

H 1 −1−1 1

Player 1 T −1 11 −1

It is a zero sum game because u1 +u2 = 0 throughout the bimatrix.



Rock Paper Scissors

Two children want to determine who uses a toy first.They play a game in which the winner goes first.Their common strategy set is {R,P,S} = {rock, paper, scissors},which can be indicated by an appropriate hand gesture.The rules specify that, if they choose different strategies, then:

1. paper wraps rock;

2. scissors cut paper

3. rock blunts scissors.

If they choose the same strategy,then the game is drawn and should be repeated.We assign a payoff of +1 to winning, −1 to losing, 0 to drawing.



RPS in Bimatrix Form

The 3× 3 payoff bimatrix is

Player 2

R P S

R 0 −1 10 1 −1

Player 1 P 1 0 −1−1 0 1

S −1 1 01 −1 0



Best Responses

In a single-person decision problem, once you understandall the actions, consequences and preferences,then you can simply choose your best, or optimal action.

What makes a game so different is that your optimal decisiondepends not only on the structure of the game,but also on what other players choose, in general.

The BoS game is a clear example:each player’s best choice (B or S)depends on what the other will do.

This demonstrates a key idea:in a game, a player’s optimal strategymust be a best response to the opponents’ strategies.



DefinitionNotation When we are focusing on player i ,we write s−i = 〈sj〉j∈N\{i} for the profileof all the other players’ strategies in the game,and S−i =

∏j∈N\{i} Sj for the set of all such profiles.

Formally:

DefinitionPlayer i ’s strategy si ∈ Si is a best response (or BR)to the other players’ strategies s−i ∈ S−iif ui (si , s−i ) ≥ ui (s

′i , s−i ) for all s ′i ∈ Si .

This definition is the key to understandingthe notion of rational strategic behavior.It is important enough to emphasize as a conclusion:

ConclusionA rational player who believes that the others are playingsome s−i ∈ S−i will always choose a best response to s−i .



Bimatrix Games

In two person bimatrix games,a relatively simple procedure finds both players’ best responses.

Step 1: For every column, find row player 1’s highestpayoff entries (often there is only one).By definition, these must be in the rowsthat are row player 1’s best responsesto the particular column being considered.Mark 1’s payoff in these entries.

Step 2: For every row, find column player 2’s highestpayoff entries (often there is only one).By definition, these must be in the columnsthat are column player 2’s best responsesto the particular row being considered.Mark 2’s payoffs in these entries.



Nash Equilibrium

In a two-person game, a Nash Equilibriumis a strategy pair (s∗1 , s

∗2 ) ∈ S1 × S2 at which:

1. s∗1 is a best response by player 1 to s∗2 ;

2. s∗2 is a best response by player 2 to s∗1 .

That is, both players’ payoffs at (s∗1 , s∗2 )

must be marked as best responses.



Prisoner’s Dilemma

Using stars to mark payoffs corresponding to best responses:

Player 2

M F

M −1 −9−1 0∗

Player 1 F 0∗ −6∗−9 −6∗

Each player has F as the only possible best response.

The only Nash equilibrium occurs at (F ,F ).



BoS

Player 2

B S

B 2∗ 01∗ 0

Player 1 S 0 1∗0 2∗

Each player’s best response is to go to the same place as the other.

There are two Nash equilibria at (B,B) and (S ,S).



Matching Pennies

Player 2

H T

H 1∗ −1−1 1∗

Player 1 T −1 1∗1∗ −1

Each player wants to “bluff”and would like to deceive the other player.

There are no Nash equilibria at all.



Rock Paper Scissors

Player 2

R P S

R 0 −1 1∗0 1∗ −1

Player 1 P 1∗ 0 −1−1 0 1∗

S −1 1∗ 01∗ −1 0

Again, each player wants to “bluff”and would like to deceive the other player.

Again, there are no Nash equilibria at all.



Summary Part I

• A game in normal or strategic form involves:(i) a list of players; (ii) their strategy sets;(iii) their payoffs, as functions of the strategy profile.

• A two-person game with finite strategy setscan be represented by a bimatrix with:(i) rows corresponding to the first player’s strategies;(ii) columns corresponding to the second player’s strategies;(iii) entries which are pairsconsisting of the two players’ respective payoffs.

• Four notable examples of bimatrix games areprisoner’s dilemma, battle of the sexes, matching pennies,and rock/paper/scissors.



Summary Part II

• In a bimatrix game, the first player’s best responseto any column chosen by the second playercan be marked with a star,as can the second player’s best responseto any row chosen by the first player.

• In a bimatrix game, any entry where both payoffs are markedis a Nash equilibriumat which the two players’ best responses coincide.

• A bimatrix game could have:a unique Nash equilibrium (like prisoner’s dilemma);or multiple Nash equilibria (like BoS);or no Nash equilibria at all(like matching pennies and rock/paper/scissors).


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