2 players Zero-Sum games</li></ul>Nash equilibrium<br />Coordination games<br />Applications of game theory<br />
What is game theory?<br />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.<br /><ul><li>In strategic games, agents choose strategies that will maximize their return, given the strategies the other agents choose.
Finite & Infinite Strategies</li></li></ul><li>Pure Strategies<br /><ul><li>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. </li></li></ul><li>An Example<br />
Mixed strategy<br />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.<br />
2-players Zero-Sum games<br />Penny Matching: <br /><ul><li> Each of the two players has a penny.
Two players must simultaneously choose whether </li></ul> to show the Head or the Tail. <br /><ul><li> Both players know the following rules:</li></ul> -If two pennies match (both heads or both <br /> tails) then player 2 wins player 1’s penny. <br /> -Otherwise, player 1 wins player 2’s penny.<br />Player 2<br />Player 1<br />
Prisoner’s dilemma<br /><ul><li>No communication:</li></ul> - Strategies must be undertaken without <br /> the full knowledge of what the other <br /> players (prisoners) will do.<br /><ul><li> Players (prisoners) develop dominant strategies but are not necessarily the best one.</li></li></ul><li>Payoff matrix for prisoner’s dilemma<br />Jane <br />Bob<br />
Equilibrium <br />Strategic equilibrium refers to the “solution” of a game: A state which a game will tend towards<br />The Prisoner’s Dilemma has one (unique Nash) equilibrium (rat out-rat out)<br />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.”<br />
Nash’s Equilibrium<br />A Nash equilibrium is a situation in which none of them have dominant Strategy and each player makes his or her best response<br />(S, T) is Nash equilibrium if S is the best strategy to T and T is the best strategy to S<br />John Nash shared the 1994 Nobel prize in Economics for developing this idea!<br />This equilibrium occurs when each player’s strategy is optimal, knowing the strategy's of the other players.<br />
Coordination games<br />coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies.<br />situations in which all parties can realize mutual gains, but only by making mutually consistent decisions<br />
Examples <br />Models the strategic conflict when two players <br />have to choose their priorities<br />Pure coordination game<br />
Exp2 –Stag hunt<br />A situation in which both players (hunters) can benefit if they cooperate<br />(hunting a stag). However, cooperation might fail, because each hunter has an alternative which is safer because it<br />does not require cooperation to succeed (hunting a hare).<br />
Applications of Game Theory<br /><ul><li>Psychology