Game Theory The Problem of Strategic Behavior What I do depends on what he does and .. ·Vice versa
Game Theory • The Problem of Strategic Behavior • What I do depends on what he does and … • Vice versa
What Von Neumann Was Trying to Do A general solution to strategic behavior how each player should play And will play,being rational And assuming the other players are .A solution that would cover ·Economics ·Politics ·foreign policy ·poker,.. But what he actually did was
What Von Neumann Was Trying to Do • A general solution to strategic behavior • how each player should play • And will play, being rational • And assuming the other players are • A solution that would cover • Economics • Politics • foreign policy • poker, … •But what he actually did was
Two Player Fixed Sum Game Fixed Sum:What helps me hurts you Strategy:A full description of what I will do in any situation Including "flip a coin,if heads do A,if tails do B" Consider "scissors paper stone"where being predictable loses Solution concept:A pair of strategies such that each is best against the other Does not include the benefit of stealing candy from babies Von Neumann demonstrated how to find the solution for any such game Provided,of course,that you have unlimited computing power to do it with
Two Player Fixed Sum Game • Fixed Sum: What helps me hurts you • Strategy: A full description of what I will do in any situation • Including “flip a coin, if heads do A, if tails do B” • Consider ”scissors paper stone” where being predictable loses • Solution concept: A pair of strategies • such that each is best against the other • Does not include the benefit of stealing candy from babies • Von Neumann demonstrated how to find the solution for any such game • Provided, of course, that you have unlimited computing power to do it with
Scissors Paper Stone 。The solution: Roll a die out of sight of your opponent .1-2 scissors,3-4 paper,5-6 stone Whatever your strategy,I win 1/3rd,lose 1/3rd,tie 1/3rd ·Average payout zero If you follow the same strategy,whatever I do gets the same average payout So a Von Neumann solution And it does not matter if you know my strategy As long as you can't see the die And similarly if I know yours Which is true in general of a VN solution
Scissors Paper Stone • The solution: • Roll a die out of sight of your opponent • 1-2 scissors, 3-4 paper, 5-6 stone • Whatever your strategy, I win 1/3rd, lose 1/3rd, tie 1/3rd • Average payout zero • If you follow the same strategy, whatever I do gets the same average payout • So a Von Neumann solution • And it does not matter if you know my strategy • As long as you can’t see the die • And similarly if I know yours • Which is true in general of a VN solution
Many Player Not Fixed Sum VN Solution concept:A set of outcomes (who gets what) Such that any outcome not in the set is dominated by one in the set Where one outcome is dominated by another if The people who prefer it(get more in it) Are sufficient,working together,to get it There may be many different solutions Each containing many outcomes So a "solution"in a very weak sense
Many Player Not Fixed Sum • VN Solution concept: A set of outcomes (who gets what) • Such that any outcome not in the set is dominated by one in the set • Where one outcome is dominated by another if • The people who prefer it (get more in it) • Are sufficient, working together, to get it • There may be many different solutions • Each containing many outcomes • So a “solution” in a very weak sense