Games of Strategy (Game Theory) Topic 1 – Part IV

Mixed Strategies Mixed-Strategies: probability mixtures of (pure) strategies.There is potentially much to be gained by using an unexpected strategy against an opponent (be unpredictable) –Mixed-strategies tell the players to use each of their pure strategies a certain percentage of the time. They are specific method of randomization –The need for randomized moves in the play of a game may arise when one player prefers a coincidence of actions, while his rival prefers to avoid it

Mixed Strategies (cont.) Expected payoff of a strategy in case of opponent’s mixed strategy: probability-weighted payoff the player can expect to receive –Sum of payoffs from the different actions, weighted by the probabilities of the opponent’s actions being taken

Mixed-Strategy Equilibrium Mixed-Strategy equilibrium: a situation in which both players choose mixed strategies that maximize their expected payoffs (no profitable deviation)

Mixed-Strategy Equilibrium (cont.) Lesson from the absence of (pure) strategy Nash equilibrium in a game: what is important in games of this type is not what players should do, but what players should not do –Neither player should always or systematically pick the same shot when faced with a similar situation –If either player engages in any determinate behavior of that type, the other can take advantage of it –The most reasonable thing for players to do here is to act somewhat unsystematically hoping for the element of surprise on defeating the opponents

An Example: The Point-Tennis Game The Point-Tennis Game: a game that analyzes the strategic behavior of two tennis players with the goal of obtain a single point in a tennis match Players: Venus and Serena Williams

An Example: The Point-Tennis Game (cont.) Serena at the net has just volleyed a ball to Venus on the baseline, and Venus is about to attempt a passing shot –Venus can try to send the ball either down the line (DL, a hard, straight shot) or crosscourt (CC; a softer, diagonal shot). –Serena must likewise prepare to cover one side or the other –Each player is aware that she must not give any indication of her planned action to her opponent, knowing that such information would be against her. Serena would move to cover the side to which Venus is planning to hit, or Venus would hit to the side Serena is not planning to cover

The Strategies If we suppose both are equally good at concealing their intentions until the last possible moment, then their actions are effectively simultaneous and we can analyze the point as a two-player simultaneous-move game The strategies: –For Venus: down-the-line passing shot; cross-court passing shot –For Serena: cover down-the-line; cover cross-court

The Payoffs The payoffs in this tennis-point game will be the fraction of times a player wins the point in any particular combination of passing shot and covering play –Assume Venus is more effective with a crosscourt passing shot than with a down-the-line passing shot when Serena covers the wrong side. But that Venus’ down-the-line passing shot is more effective than her crosscourt shot when Serena covers the right side –Consider the fact that that Venus (and any other player) is more likely to win the point when Serena moves to cover the wrong side of the court –Now, we can work a reasonable set of payoffs

The Payoffs (cont.) Venus will be successful with a down-the-line passing shot 80% of the time if Serena covers crosscourt Venus will be successful with a down-the-line passing shot only 50% of the time if Serena covers down-the-line Venus will be successful with a crosscourt passing shot 90% of the time if Serena covers down-the-line Venus will be successful with a crosscourt passing shot only 20% of the time if Serena covers crosscourt

The Payoffs (cont.) Clearly, the fraction of times Serena wins this tennis point is just the difference between 100% and the fraction of time that Venus wins Then payoffs are as follows: –If strategies are (DL, DL) : (50, 50) –If strategies are (DL, CC): (80,20) –If strategies are (CC, DL): (90,10) –If strategies are (CC, CC): (20, 80)

How to Find Mixed-Strategy Equilibria “Keep-the-opponent-indifferent” method (KTOI): randomize in such a way that your partner will be indifferent between her pure strategies (i.e., make your opponent’s expected payoffs from both strategies to be the same) –A rational player will randomize between her possible strategies ONLY if she gets an equal expected payoff from both strategies. Otherwise, she will choose the (pure) strategy with the highest payoff Every simultaneous-move game has a Nash equilibrium in mixed strategies