09.DynamicAndSimultaneous

# 09.DynamicAndSimultaneous - DYNAMIC GAMES INVOLVING...

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DYNAMIC GAMES INVOLVING SIMULTANEOUS MOVES (or: Dynamic Games with Imperfect Information) Example: The state, the inspector and the thief. The state (Player 1) first makes a choice between offering a reward of R > 0 TL and 0 TL to the inspector for successfully detecting the thief. Next, the inspector and the thief play a simultaneous-move game where the inspector chooses between inspect ” and “ don’t inspect ,” and the thief chooses between rob ” and “ don’t rob .” State R “0” Subgame A Subgame B How can we extend the idea of backward induction to this type of games? We proceed as follows… 1. We identify the portions of the overall game that can be analyzed separately… These portions are called “SUBGAMES”. 2. We identify the Nash equilibria of these “subgames”, then proceed backward toward the origin of the overall game, in accordance with the logic of backward induction. Thief Inspector rob don’t inspect 150, 0 , 5 250, -5 , 0 don’t 0, 5 , 20 200, 3 , 0 Thief Inspector Rob don’t inspect 100, 10 , 5 200, 0 , 0 don’t 0, 5 , 20 200, 3 , 0 1

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In this game, Subgame A… The Nash Equilibrium is {Inspect, Rob} Subgame B… The Nash Equilibrium is {Don’t inspect, Rob} The state should expect that the strategies of inspector and the thief (as rational players) be best replies to each other (that is, strategies that are Nash equilibria). Therefore in Subgame A the state should expect the payoff “100,” and in Subgame B the state should expect the payoff “0.” Reducing the game to… State R “0” (100, 10 , 5) ( 0, 5 , 20) The state should choose “R”, which leads to the following strategies. . State … choose “R” Inspector … “Inspect in subgame A, don’t inspect in subgame B” Thief … “rob in both subgames” STRATEGIC-FORM REPRESENTATION OF DYNAMIC GAMES… (with two players) We identify the strategy sets of the players in the Dynamic game and then form the game matrix according to these strategy sets. Using the strategic form helps us finding the Nash equilibria. 2
Example 1. player 1 out in (2, 2) player 2 smash smooth (0,0) (4, 1) Backward induction solution is {“ in ”, “ smooth if in ”} Strategy sets are: Player 1… {out , in} Player 2… {smash if in, smooth if in} Claim .. {in, smooth if in} is a Nash Equilibrium. BUT…. {out, smash if in} is also a Nash Equilibrium! Payoffs, (2, 2) Nash equilibria of Dynamic games are best seen in their Strategic forms… Player 2 Player 1 Smash if in Smooth if in Out (2 , 2) (2 , 2) In (0 , 0) (4 , 1) (Out, Smash if in) IS NOT REASONABLE… It involves an empty threat by Player 2. The threat is “ empty ” because playing “smash if in” is not in Player 2’s self interest . Backward induction solution eliminates this outcome… 3

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SUBGAME ANALYSIS IN THIS EXAMPLE player 1 out in SUBGAME (2, 2) player 2 smash smooth (0,0) (4, 1) In the subgame extending after player 1 played “in,” only player 2 moves. In this subgame player 2 should play “smooth” and get
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09.DynamicAndSimultaneous - DYNAMIC GAMES INVOLVING...

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