Intro to Game Theory

Revisiting the territory we have covered

A look at the skeleton

• There is a fairly small set of ideas which we have seen developed with a rich variety of examples.

• Today we look at this skeleton. The text develops the examples and we have discussed a large number of them.

• My advice for study. Read the assigned text and readings carefully. Work problems. Try especially to understand the assigned problems and the worked out examples in the text.

What is a strategic game?

• Interacting players• Set of possible actions for each player• Action profiles• Preferences over action profiles

Example: The Stag Hunt

2 , 2 0 , 1

1 , 0 1 , 1

StagStag Hare

HarePlayer 1

Player 2

Actions Possible?

Action Profiles?

Preferences?

Dominating strategies.

• Action A strictly dominates action B for a player if he prefers the outcome from doing A to that from doing B, no matter what action the other player takes.

• Does either strategy in the Stag Hunt dominate the other strategy?

Another game

1 0 , 10 0, 11

1 1, 0 1, 1

Strategy AStrategy A Strategy B

Strategy BPlayer 1

Player 2

Does either strategy strictly dominate the other forPlayer 1? Does either strategy strictly dominate the other forPlayer 2? What are games like this called?

WeaklyDominating strategies.

• Action A weakly dominates action B for a player if he likes the outcome from doing A to at least as well as that from doing B, no matter what action the other player takes.

• And for some actions by the other player, he likes the outcome from A better.

How about this one?

1 0 , 10 0, 10

1 0, 0 1, 1

Strategy AStrategy A Strategy B

Strategy BPlayer 1

Player 2

Does either strategy weakly dominate the other for Player 1? Does either strategy strictly dominate the other forPlayer 1?

Best response functions and Nash Equilibrium

• The best response function for any player i, is a function that maps the list of actions by other players into the list of actions that are best responses to what the others did.– Sometimes there is only one best response.

• A Nash equilibrium is a set of actions by the players such that each player’s action is a best response to the actions of the other players.

Example: The Stag Hunt

2 , 2 0 , 1

1 , 0 1 , 1

StagStag Hare

HarePlayer 1

Player 2

B1(Stag)= {Stag} B1(Hare)={Hare}

B2(Stag)={Stag} B2(Hare)={Hare}

Nash equilibrium for 2-player game

• For a two-player game, a Nash equilibrium consists of an action for each player such that each player’s action is a best response to the other player’s action.

• Method of stars works for games with finite number of strategies.

• For game with continuum of strategies (e.g. Cournot equilibrium, one calculates best response function for each player, which typically gives you two equations in two unknowns, which you then solve.

Nash equlibrium for this game?

1 0 , 10 0, 10

1 0, 0 1, 1

Strategy AStrategy A Strategy B

Strategy BPlayer 1

Player 2

Nash equilibria for this game?

1 0 , 10 0, 11

1 1, 0 1, 1

Strategy AStrategy A Strategy B

Strategy BPlayer 1

Player 2

What can you say about a Nash equilibrium in a game where each player has a “strictly dominant strategy”?What do the best response functions look like if there is a strictly dominant strategy?

Games with more than 2 players

• Each player’s best response can in general depend on actions of all other players.

• A Nash equilibrium is a list of one action by each player, such that each player’s action is a best response to the actions of the other players.

Mixed strategies

• Possible strategies include randomizing between pure strategies.

• For this theory, we need to specify von Neumann Morgenstern utilities.

• Players seek to maximize Bernoulli payoff function which is the expected value of von Neumann-Morgenstern utility.

• Here the intensity of preference as well as order of preference matters.

Risk aversion, risk neutrality, risk loving

• With money prizes, von Neumann Morgenstern preferences are given by a function u(y) of prize amount y.

• Risk neutral if u’(y) is constant.• Risk averse if u’( y) is decreasing.• Risk loving if u’(y) is increasing.

Mixed Strategy Nash equilibrium

• A mixed strategy Nash equilibrium is a list of mixed strategies for each player, such that each player’s mixed strategy is a best response to the other players’ mixed strategies. (Mixed strategies are defined to include the pure strategies as special cases.)

• Example: Matching pennies Draw equilibrium diagram. Change payoffs.

Extensive game with perfect information?

• Set of players• Set of terminal histories-a full history of a

game –possible courses of the game• Player function: whose turn it is at each point

in the game• Payoffs: to each player from each possible

terminal history

The entry game

Two players: Challenger and Incumbent.Challenger moves first. Challenger either enters the contest or stays out.If challenger stays out, game ends. If challenger gets payoff 1 and incumbent gets 2.If Challenger enters, it is incumbent’s turn. Either he yields or he fights. If he yields, challenger gets 2, incumbent gets 1. If he fights,Both get 0.

In the entry game

What are the possible strategies for the entrant? What are the possible strategies for the incumbent? What are the Nash equilibria?What are the subgame perfect equilibria (um?)What does this say about ``credible threats?’’

Coalitional Games

• Focus on what groups can accomplish if they work together.

• Contrast to Nash equilibrium which focuses on what individuals can do acting alone. (sometimes known as non-cooperative game theory)

Coalitional Game with transferable payoffs

• A set of players N.• A coalition S is a subset of N.– Grand coalition is N itself.

• Coalitional game with transferable payoffs assigns a value v(S) to every subset of S.

• An action for the coalition S is a distribution ofIts total value to its members.

Think of v(S) as an amount of “money” that the coalition can earn on its own and can divide this money in any way that adds to v(S).

The Core

• The core of a coalitional game is the set of outcomes x (actions by the grand coalition) such that no coalition has an action that all of its members prefer to x.

A game with transferable payoffs and no core: Majority redistribution game

• Players 1, 2, and 3. • Non-empty subsets of N={1,2,3} are N, {12}, {13}, {23},{1},{2},{3}.• A cake whose total value is 1 is to be divided. Any

coalition that is a majority can choose how to divide it.• Then v({12})=v({23})=v({13})=v({123})=1 and

v({1})=v({2})=v({3})=0.• Actions available to any coalition are possible divisions

of the cake. For example, coalition{12} can choose any division such that x1≥0, x2≥0, and x1+x2=1.

Coalitional Games without transferable payoffs

• Roommate Assignment• Marriage assignment• College admissions

• What are the possible coalitions? What are the payoffs?