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NashTrigger

# NashTrigger - – Play b in all periods as long as player 2...

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Notes on Nash Trigger Strategies Daniel Egel November 10, 2008 1 Nash Trigger Strategy Lets take a look at one of our familiar examples: Player 1 Player 2 i j k l a 1,2 2,4 3,1 5,3 b 2,1 3,3 4,2 4,4 c 1,2 3,2 1,1 4,2 Now if only Nash threat points are available that means that player 1 would punish by playing { c } and player 2 would punish by playing { j } . Thus an individually rational play in this setup would have to beat this point. Note the following: The stage game Nash is { c;j } . Thus there are only two individually rational plays: { b;l } , { a;l } Question: Can the individually rational stage-game outcome ( b,j ) be supported by Nash trigger strategies? The answer is no since it is not individually rational. 1 Lets consider another point, say { b;l } . From class we learned that any individually rational play can “support” a SPE. Lets show that this is true for this point. Assume a discount rate of δ = 0 . 9 and a game with inﬁnite periods. A Nash trigger strategy for these two players would be Player 1:

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Unformatted text preview: – Play { b } in all periods as long as player 2 has always played { l } – Play { c } if player 2 has ever played something other than { l } Player 2: – Play { l } in all periods as long as player 2 has always played { b } – Play { j } if player 2 has ever played something other than { b } • So the ﬁrst question is who might have an incentive to deviate from the cooperation point. Clearly it is player 1. • Now let see if our strategy can convince player 1 not to deviate: 1 Hence the confusion in class. 1 – If he cooperates he gets 4 1-δ = 40 – If he defects he gets 5 + 3 δ 1-δ = 32 – So clearly he will not want to defect • Is it ex-post rational? In other words, will player 1 have any incentive to deviate from this strategy? – Given that they are playing at the threat point, { c;j } neither player will have an incentive to deviate – Thus it is a SPE 2...
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NashTrigger - – Play b in all periods as long as player 2...

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