In: Economics
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.