Simultaneous Move Games Decision making without knowledge of the strategy choice of opponents Simultaneous Moves Arise when players have to make their strategy choices simultaneously without knowing the strategies that have been simultaneously chosen by the other player s Student studies for a test the teacher writes questions Two firms f independently d d l d decide d whether h h or not to develop d l and d market k a new product While there is no information about what other players will actually choose we assume that the strategic choices available to each player are known by all players Players must think not only about their own best strategic choice but also the best strategic choice of the other player s We will consider both discrete and continuous strategy spaces Normal or Strategic g Form A simultaneous move game is depicted in Normal or Strategic g form usingg a ggame table that relates the strategic choices of the players to their payoffs The convention is that the row player s payoff is listed fi t and first d the th column l player s l payoff ff iis listed li t d second d Column Player Row R Player Strategy R1 SStrategy C1 a b SStrategy C2 c d Strategy R2 e f g h For example if Row player chooses R2 and Column player chooses C1 the Row player s payoff is e and the Column player s payoff is f Special p Zero Sum Form For zero or constant sum games knowing the payoffs sum to zero or some other constant allows us to write a simultaneous py move ggame in normal form more simply Warden Pi Prisoner Cli b Wall Climb W ll Guard Wall 1 1 Inspect Cells 1 Dig Tunnel 1 1 Payoffs are shown only for the Prisoner the Warden s payoffs are the negative of the prisoner ss payoff prisoner The Role of Beliefs When players move simultaneously what does it mean to say that in equilibrium strategies are a mutual best response You cannot see what the other is doing and condition your behavior on their move In simultaneous move games rational players consider all of the strategies their opponents may take and they j p form beliefs subjective probabilities about the likelihood of each strategy their opponent s could take After forming these beliefs beliefs rational players maximize their expected payoff by choosing the strategy that is a best response to their beliefs about the play of their opponent s opponent s The same is true of the opponent s opponent s Coordination Game Example How would yyou p playy this game g Example p of the Role of Beliefs Consider the pure coordination game Column Player Row Player X X 0 0 Y 1 1 Y 1 1 0 0 Suppose Row player assigns probability p p 5 5 to column player playing Y Then Row s best response to this belief is to play X Row s expected payoff from playing X is 0 1 p 1 p p while Row s expected payoff from playing Y is 1 1 p 0 p 1 p Since we assumed p 5 the expected payoff to Row from playing X p is greater than h the h expected payoff ff to Y 1 p How Might g Such Beliefs be Formed Players subjective beliefs about the play of an opponent in a simultaneous move game may be formed in one of several ways Introspection given my knowledge of the opponent s payoffs what would I do if I were the other player History repeated games only what strategy has the same opponent played in the past Imitation learning from others what strategies have players other than my current opponent chosen in this type of strategic setting Pre play communication communication Other type of signaling We focus for now on the first introspective method Pure vs Mixed Strategies g A player pursues a pure strategy if she always chooses the same strategic action out of all the strategic action choices available to her in every round e g Always Al refuse f tto clean l th the apartment t t you share h with your roommate Ap player y p pursues a mixed strategy gy if she randomizes in some manner among the strategic action choices available to her in every round e g e g Sometimes pitch a curveball curveball sometimes a slider mix it up keep them guessing We focus for now on pure strategies only Example p Battle off the Networks Suppose there are just two television networks Both are battling for shares of viewers 0 100 Higher shares are preferred higher advertising revenues revenues Network 1 has an advantage in sitcoms If it runs a sitcom it always gets a higher share than if it runs a game show Network 2 has an advantage in game shows If it runs a game show it always gets a higher share than if it runs a sitcom Networkk 2 Network 1 Sitcom Sitcom 55 45 55 Game Show 52 48 52 Game Show 50 50 45 55 Nash Equilibrium We cannot use rollback in a simultaneous move game so how do we find a solution We determine the best response of each player to a particular choice of strategy by the other player We do this for both players Note that in thinking of an opponent s best response we are using introspection to form beliefs about what the rational opponent will do If each player s strategy choice is a best response to the strategy choice of the other player then we have found a solution or equilibrium to the game This solution concept p is know as a Nash equilibrium q after John Nash who first proposed it A game may have 0 1 or more Nash equilibria Best Response Analysis Best response analysis a k a cell by cell inspection is the most reliable method for finding Nash equilibria First find Network 1 s best response to Network 2 s possible strategies If Network 2 runs a sitcom Network 1 s best response p is to run a sitcom Circle Network 1 s payoff in this case 55 If Network 2 runs a game show Network 1 s best response is to run a sitcom sitcom Circle Network 1 s 1 s payoff in this case case 52 Network 2 Network 1 Sitcom Sitcom 55 45 Game Show 52 48 Game Show 50 50 45 55 Best Response Analysis Continued Next we find Network 2 s best response If Network 1 runs a sitcom Network 2 s best response is to run a game show Circle Network 2 s p payoff y in this case 48 If Network 1 runs a game show Network 2 s best response is to run a game show Circle Network 2 s payoff in this case 55 The unique Nash equilibrium is for Network 1 to run a sitcom and Network 2 to run a game show This is found by the cell with the two circled payoffs This is the method of best response analysis for locating Nash …