UncertainTerminationABCabc

Consider the following game that is played with uncertain termination.That is, after each round thegamecontinueswithprobabilityandendswithprobability1,whereuponeachplayergetsapayoffof 0. Players move simultaneously and independently. Then each player is informed about the actionstaken by the other player in the round and, given this, they play it again with probability. The payofffor the whole game is the expectation of the sum of the payoffs a player obtains. You may remember aversionofthisquestionfromLesson8wherethegamewasplayedonlytwice.Notethepayofffor(A,c)haschanged.

a b c

A3,25,02,2

B2,2 4,4 1,2

C 1,2 0,2 2,3

CertainTerminationABCabc

Let'stakethegamefromaboveandmake2smallchanges...thepayoffto(B,c)haschangedto1,5and(A,c)is1,1.Inthiscase,definethetotalpayoffequaltothesumofpayofffrombothroundsbecausebothwillsurelybeplayed.

Player2

a b c

A 3,25,01,1

Player1 B 2,2 4,4 1,5

C 1,2 0,2 2,3

a)IsitpossibletohaveB,baspartofaSPNEinthefirstroundofa2-roundgame?¹Whyor

whynot?

¹hint:beginbyspecifyingthefollowing"strategy:"if(B,b)isobservedinthefirstperiod,play

(A,a);otherwiseplay(C,c)thenseewhathappensandworkfromthere.

b)IsiteverpossibletohaveB,baspartofaSPNEinthefirstroundofamulti-roundgame?

Whyorwhynot?