What is game theory? Game theory is a formal way to analyze strategic interaction among a group of rational players (or agents) Game theory has applications -Economics/Politics/Sociology/Law/Biology >The"double helix"and unifying tool for social scientists
9 What is game theory? ◼ Game theory is a formal way to analyze strategic interaction among a group of rational players (or agents) ◼ Game theory has applications ➢ Economics/Politics/Sociology/Law/Biology ➢ The “double helix” and unifying tool for social scientists
Classic Example:Prisoners'Dilemma Two suspects held in separate cells are charged with a major crime.However,there is not enough evidence. Both suspects are told the following policy: If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail. If both confess then both will be sentenced to jail for six months. >If one confesses but the other does not,then the confessor will be released but the other will be sentenced to jail for nine months. Prisoner 2 Mum Confess Mum -1 ,-1 -9， 0 Prisoner 1 Confess 0 ,-9 -6 -6 10
10 Classic Example: Prisoners’ Dilemma ◼ Two suspects held in separate cells are charged with a major crime. However, there is not enough evidence. ◼ Both suspects are told the following policy: ➢ If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail. ➢ If both confess then both will be sentenced to jail for six months. ➢ If one confesses but the other does not, then the confessor will be released but the other will be sentenced to jail for nine months. -1 , -1 -9 , 0 0 , -9 -6 , -6 Prisoner 1 Prisoner 2 Confess Mum Confess Mum
Example:The battle of the sexes At the separate workplaces,Chris and Pat must choose to attend either an opera or a prize fight in the evening. Both Chris and Pat know the following: Both would like to spend the evening together. But Chris prefers the opera. >Pat prefers the prize fight. Non-zero-sum game Pat Opera Prize Fight Opera 2 1 0 0 Chris Prize Fight 0 0 2
11 Example: The battle of the sexes ◼ At the separate workplaces, Chris and Pat must choose to attend either an opera or a prize fight in the evening. ◼ Both Chris and Pat know the following: ➢ Both would like to spend the evening together. ➢ But Chris prefers the opera. ➢ Pat prefers the prize fight. ◼ Non-zero-sum game 2 , 1 0 , 0 0 , 0 1 , 2 Chris Pat Prize Fight Opera Prize Fight Opera
Example:Matching pennies ■ 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. Zero-sum game:no way for collaboration Player 2 Head Tail Head -1 1 1 -1 Player 1 Tail -1 12
12 Example: Matching pennies ◼ 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. ➢ Zero-sum game: no way for collaboration -1 , 1 1 , -1 1 , -1 -1 , 1 Player 1 Player 2 Tail Head Tail Head
Static (or simultaneous-move)games of complete information A static (or simultaneous-move)game consists of: A set of players (at least Player 1,Player 2,... two players) Player n} For each player,a set of >SI S2 ...Sn strategies/actions ■Payoffs received by > each player for the ui(Sp,S2 ....)for all combinations of the S1∈S,S2∈S2,.Sn∈Sr strategies,or for each player,preferences over the combinations of the strategies 6
13 Static (or simultaneous-move) games of complete information ◼ A set of players (at least two players) ◼ For each player, a set of strategies/actions ◼ Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies ➢ {Player 1, Player 2, ... Player n} ➢ S1 S2 ... Sn ➢ ui (s1 , s2 , ...sn ), for all s1S1 , s2S2 , ... snSn . A static (or simultaneous-move) game consists of: