Alice and Bob are a couple, playing the infinitely repeated game with the following stage game...

 

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Alice and Bob are a couple, playing the infinitely repeated game with the following stage game and discount factor 8. Every day, simultaneously, Alice and Bob spend XÃE [0 1] and XBE [0 1] fraction of their time in their relationship, respectively, receiving the stage payoffs UA = ln (XA+XB )+1 −XA and UB= ln (XA+XB )+1-XB, respectively. (Alice and Bob are denoted by UA and UB, respectively.) For each of the strategy profiles below, find the conditions on the parameters for which the strategy profile is a subgame-perfect equilibrium. a) Both players spend all of their time in their relationship (i.e. XA = XB = 1) until somebody deviates; the deviating player spends 1 and the other player spends 0 thereafter. (Find the range of 8.) b) There are 4 states: E (namely, Engagement), M(namely, Marriage), D1, and D2. The game starts at state E, in which each player spends & E (01). If both spend E (01), they switch to state M; they remain in state E otherwise. In state M, each spends 1. They remain in state M until one of the players i spends less than 1 while the other player spends 1, in which case they switch to state Di. In state Di, player i spends and the other player spends 1-x, forever. (Find the set of inequalities that must be satisfied by the parameters 8, X, XA, XB). Alice and Bob are a couple, playing the infinitely repeated game with the following stage game and discount factor 8. Every day, simultaneously, Alice and Bob spend XÃE [0 1] and XBE [0 1] fraction of their time in their relationship, respectively, receiving the stage payoffs UA = ln (XA+XB )+1 −XA and UB= ln (XA+XB )+1-XB, respectively. (Alice and Bob are denoted by UA and UB, respectively.) For each of the strategy profiles below, find the conditions on the parameters for which the strategy profile is a subgame-perfect equilibrium. a) Both players spend all of their time in their relationship (i.e. XA = XB = 1) until somebody deviates; the deviating player spends 1 and the other player spends 0 thereafter. (Find the range of 8.) b) There are 4 states: E (namely, Engagement), M(namely, Marriage), D1, and D2. The game starts at state E, in which each player spends & E (01). If both spend E (01), they switch to state M; they remain in state E otherwise. In state M, each spends 1. They remain in state M until one of the players i spends less than 1 while the other player spends 1, in which case they switch to state Di. In state Di, player i spends and the other player spends 1-x, forever. (Find the set of inequalities that must be satisfied by the parameters 8, X, XA, XB).

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a For the strategy profile in which both players spend all of their time in the relationship until somebody deviates the conditions for subgameperfect ... View the full answer