Session 2: The Return of Game

Theory

St. Xavier’s College, Mumbai

January, 2014

Kabira Namit (kabira.namit@gmail.com)

Game Theory: The Basics•Game Theory analyses how rational decisions are made when decisions are interactive: This means that each individual’s best choice depends on the choices made by other individuals.•Thus in the cases discussed by game theory, each individual’s perception or belief about what others might do has a direct impact on that person’s choice.

Game Theory vs. Conventional Economics•In the conventional model of the market economy, each individual’s best choice depends only on that individual’s preferences and costs, it does not depend on the choices made by others•Thus, game theory models give us very different results from market economic theory where individual maximization always results in social maximization

Continued•In game theory, social maximization does not always follow from individual maximization

The Elements of Game Theory•The players: These are the individuals, i, who make decisions. Each player's goal is to maximize his/her utility by a choice of actions. •The actions of each player ai are the choices they can make. •The information set for each player is the knowledge he/she has at each stage of the game of the values of different variables.

Continued•A player's strategy si are the actions player i chooses at each stage of the game, given his/her information set. •A player's payoff Πi(s1,s2,..sn) is the expected utility of the player given the strategies chosen by him/herself and all other players.

Continued•The equilibrium s*=(s1*,...,sn*) is a strategy combination consisting of a best strategy for each of the n players in a game. There are a number of different equilibrium concepts in game theory which we will discuss in turn.

The Core Components of Game Theory•A game is a set of rules known to all players according to which the players can get different payoffs by selecting different strategies•To make the modelling tractable, the rules themselves are fixed and cannot be changed by the players•However, the payoff for each player depends on the choices of other players

Examples of Games: Games of Pure Competition•Players have exactly opposed interests•Their payoffs are complements of each other•For all action profiles,

•a ∈ A, u1(a) + u2(a) = c•Special case: Zero sum game•We need to think about the utility of only one player

Continued•One player wants to match, the other wants to mismatch

Heads Tails

Heads 1, - 1 -1, 1

Tails -1, 1 1, - 1

A chance at glory

Games of Coordination•Players have exactly the same interests•All players want the same thing•a ∈ A, u1(a) = u2(a) •We often write such games with a single payoff per cell

Continued•Lets not crash

Left Right

Left 1, 1 - 10, - 10

Right - 10, - 10 1, 1

Extensive Form of Coordination Game

B

A

B

1,1-10,-101,1 -10,-10

•Non – simultaneous

•Similar to chess where the player playing black responds to the player playing white

•Tree diagram

Battle of the Sexes

Wife wants to see…

Husband wants to see…

Continued•The most interesting games combine elements of cooperation and competition

Wolf of Wall Street

Joe B Carvalho

Wolf of Wall Street

2, 1 0, 0

Joe B Carvalho

0, 0 1, 2

Equilibrium Concepts in Game Theory•A number of equilibrium concepts are used in game theory. The best known are the Dominant Strategy and Nash Equilibrium concepts. •In discussing equilibria we use the notation s-i to denote the vector of strategies of all players other than i. •Player i's best response to the strategies s-i chosen by other players is the strategy s* that yields the greatest payoff

Continued•Mathematically, this is denoted as:π i(si*,s-i) ≥ π i(si', s-i) ▼si' ≠ si* •The strategy si* is a dominant strategy if it is a player's strictly best response to any strategies the other players may pick, in the sense that whatever strategies they pick, his payoff is highest with si* πi(si*,s-i) ≥ πi(si',s-i) for ▼s-i, and ▼si' ≠ si*

Concept of Nash Equilibrium•The strategy combination s* is a Nash Equilibrium if no player has an incentive to deviate from his strategy given that the other players do not deviate, πi(si*,s-i*) ≥ πi(si',s-i*)

Continued•Note that the definition of Nash equilibrium lacks the extra ▼s-i of the dominant strategy equilibrium so a Nash strategy only needs to be the best response to other Nash strategies, not to all possible strategies. This is also shown in the definition by the fact that the Nash maximizes πi(si*,s-i*) and not πi(si*,s-i).

Continued•Thus the Nash is a best response on the assumption that other players are playing Nash. •Thus, Varian argues that the Nash equilibrium is an equilibrium in actions and beliefs.

Nash Equilibrium•In the Nash equilibrium each person asks:•If I follow a particular strategy, assuming that others are following a particular strategy, will they continue to follow their particular strategies if they know that I am following this strategy?•If the answer to this question is yes for every individual following a strategy, the combination of strategies for all individuals is a Nash equilibrium for the game

Continued•The characteristic of a Nash equilibrium is that once everyone is playing a Nash, no-one will play any other strategy, even if the Nash does not appear rational for all players collectively•There may be more than 1 Nash equilibrium for a game, and the outcome may be socially undesirable because with more than one equilibrium, rational players can make mistakes and end up with suboptimal outcomes

Identifying Nash Equilibrium

Take each strategy combination in turn For example, take (Left, Right), and ask (i) will A play Left if B plays Right and (ii) will B play Right if A plays Left If the answers to both questions is yes, then (Left, Right) is a Nash equilibrium

Left (B) Right (B)

Left (A) 5,1 4,4

Right (A) 9, -1 0,0

Battle of the Sexes

Wolf of Wall Street

Joe B Carvalho

Wolf of Wall Street

2, 1 0, 0

Joe B Carvalho

0, 0 1, 2

Coordination game

Left Right

Left 1, 1 - 10, - 10

Right - 10, - 10 1, 1

Game of Matching Pennies

Heads Tails

Heads 1, - 1 -1, 1

Tails -1, 1 1, - 1

Justifications for a Nash Equilibrium•It gives a formal definition of equilibrium if there is a unique self-evident way to play a game. If there is an equilibrium to a game, it must be because rational players have no incentive to move away from it.•The Nash may represent a self-enforcing convention. If the players can engage in (non-binding) communication before the game then they will have to work out a strategy combination which is Nash.

Problems with Nash Equilibrium•There may be more than one possible Nash equilibrium. Game theory does not tell us how equilibria are selected.

•The Nash equilibrium solution may be so complex that it is difficult to argue that the solution is the self-evident equilibrium which would be chosen by rational actors.

•The Nash simply describes the likely conventions which emerge. It does not explain the emergence of beliefs about what other agents will do.

Dominant Strategy Equilibrium•Some games (like the prisoner’s dilemma) have a stronger equilibrium called the Dominant Strategy equilibrium

•Here, each player asks if there is a strategy for that player that is the best strategy regardless of the particular strategies played by the other players.

•If it exists, such a strategy is called a dominant strategy for that player

Continued•If all players have a dominant strategy, the game has a dominant strategy equilibrium, and these equilibria are very stable

•All dominant strategy equilibria are also Nash equilibria, but not vice versa