Question

In: Economics

explain why a nash equilibrum in a dynamic game may not be subgame perfect? using example...

explain why a nash equilibrum in a dynamic game may not be subgame perfect? using example
show how non credible theats are ruled out

Solutions

Expert Solution

QUESTION
ans .
A Nash equillibrium is a combination of strategies for all players actual strategy , which means that each player , acting in isolation cannot achive a better outcome for themselves by altering their strategy, given their strategy each other player has adopted.
A sub game perfect nash equillibrium is a Nash equillibrium is a Nash equillibrium with the additional restriction that each individual decision in a players strategy would be the one that gets them the best outcome.

It can be diffficult to check whether a strategy profile is a subgame - perfect equllibrium in infinite horizons or dynamic games , where some paths can go forever without ending the game .There is however a simple technique that can be used to which is called single deviation principle .

let's take an example in a game there may be histories where all the previous actions are known but the players may move simultaneously . such histories are called stages .  a player everyday plays a battle of sexes , knowing what each player has played in each previos days, we have a stage at which players move simultaneously and a new game starts . or consider an example of bargaining where two people or more have a differed opinion after a billion rounds.
SINGLE DEVIATION TEST
consider a strategy profile s* . Pick any stage ( after any history of moves ). Assume that we are at that stage .Pick also a player i* who moves at that stage .fix all the other players ' moves as predcribed by the strategy profie s* at the current stage as well as in the following game . fix also the moves of player i* at alll the future dates , but let his moves at current stage that gives a higher payoff than s* fails the single deviaton test at an stage fpr player i* then s* cannot be a subgame perfect equillibrium . this is because s* does not lead to a Nash equillibrium at the subgame that starts at that stage , as player i* has an incentive to deviate to the strategy according to which i* plays the better moves at the current stage but follows s*i in the remainder of the subgame .it turns out that in a multistage game that is co tnious at infinity, the converse is also true . if s* passes the single deviation principle at every stage (after every history of previous moves ) for every player, then it is a subgame - perfect equillibrium .

PRINCIPLE AND BRIEF
In a multistage game that is continuous at infinity , a strategy profile i s asubgame perfect .Nash equillibrium if and only if it passes the single deviation test at every stage for every player.

Non credible credits are ruled out as follows:
A non credible threat is aterm used in game theory and econmics to describe a threat in a sequential game that arational player would actually not carry out , because i would not be in his interset to do so.
EXAMPLE.
A carried a bomb and waks up to a person B .Atells B he will set off the bomb killing them both ,unless B gives him all his money .If A is rational and non suicidal he stands nothing to gain from setting of the bomb ,so his threatcannot be considered credible on the otherhand , a person in the situation of B right might give A his money ,fearing that A is not rational ,or might even be suicidal .
Those Nash equillibria that rely on Non credible threats can be eliminated through backward induction the remaining eqillibria are called subgame perfect Nash eqillibria.




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