Nash equilibrium Coordination games Applications of game theory
What is game theory? Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology (most notably evolutionary biology and ecology), engineering, political science, international relations, computer science, and philosophy.
In strategic games, agents choose strategies that will maximize their return, given the strategies the other agents choose.
A design tool.
The mathematics of human interactions.
A promise for the unification of social sciences.
History of game theory
von Neumann wrote a key paper in 1928
1944: “Theory of Games and Economic Behavior” by von Neumann and Morgenstern
1950: Nash invents concept of Nash equilibrium
Game theory booms after this…
1994: Harsanyi, Nash, and Selten win Nobel Prize in economics for game theory work
Key elements of a game
Players: Who is interacting?
Strategies: What are their options?
Payoffs: What are their incentives?
Information: What do they know?
Rationality: How do they think?
Types of game
Cooperative or non-cooperative
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Finite & Infinite Strategies
The upper value of the game is equal to the minimum of the maximum values in the columns.
The lower value of the game is equal to the maximum of the minimum values in the rows.
Mixed strategy A mixed strategy game exists when there is no saddle point. Each player will then optimize their expected gain by determining the percent of time to use each strategy.
2-players Zero-Sum games Penny Matching:
Each of the two players has a penny.
Two players must simultaneously choose whether
to show the Head or the Tail.
Both players know the following rules:
-If two pennies match (both heads or both tails) then player 2 wins player 1’s penny. -Otherwise, player 1 wins player 2’s penny. Player 2 Player 1
- Strategies must be undertaken without the full knowledge of what the other players (prisoners) will do.
Players (prisoners) develop dominant strategies but are not necessarily the best one.
Payoff matrix for prisoner’s dilemma Jane Bob
Equilibrium Strategic equilibrium refers to the “solution” of a game: A state which a game will tend towards The Prisoner’s Dilemma has one (unique Nash) equilibrium (rat out-rat out) No one player can unilaterally change his strategy for a better outcome: ”I can do no better, given that the other player keeps doing what he is doing.”
Nash’s Equilibrium A Nash equilibrium is a situation in which none of them have dominant Strategy and each player makes his or her best response (S, T) is Nash equilibrium if S is the best strategy to T and T is the best strategy to S John Nash shared the 1994 Nobel prize in Economics for developing this idea! This equilibrium occurs when each player’s strategy is optimal, knowing the strategy's of the other players.
Coordination games coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies. situations in which all parties can realize mutual gains, but only by making mutually consistent decisions
Examples Models the strategic conflict when two players have to choose their priorities Pure coordination game
Exp2 –Stag hunt A situation in which both players (hunters) can benefit if they cooperate (hunting a stag). However, cooperation might fail, because each hunter has an alternative which is safer because it does not require cooperation to succeed (hunting a hare).