Question

Consider the following game with sequential moves. Consider the following game with two players and sequential moves. Rob moves first and has to choose one of the two following actions: Invest (I) or Not Invest (NI); Anne moves second, after having observed the action chosen by Rob and has to choose one of the two following actions: Reward (R) or Not Reward (NR). The payoffs of Rob are as follows: when he chooses I and Anne R: 6; when he chooses I and Anne NR: 1; when he chooses NI and Anne R; 3; when he chooses NI and Anne NR: 0. The payoffs of Anne are: when Rob chooses I and she chooses R: 2; when Rob chooses I and she chooses NR: 4; when Rob chooses NI and she chooses R; 2; when Rob chooses NI and she chooses NR: 1.

[10 marks] Use the graphical tree representation to describe the extensive form of the game.

[15 marks] Find a subgame perfect equilibrium of the game considered. Carefully describe the equilibrium strategies of the two players, as well as the equilibrium outcome.

[15 marks] Find a Nash equilibrium of the game considered that is not subgame perfect where Rob chooses I. Explain why at this equilibrium the strategy of the second player features a non-credible threat.

[10 marks] Suppose the payoff of Anne when Rob chooses NI and she chooses R is modified and is now equal to 1. In this case there is another subgame perfect equilibrium, in addition to the one you found in (b): find this other subgame perfect equilibrium, carefully describing the equilibrium strategies of the two players.

Answer 1

The subgame perfect equilibrium for the game with sequential moves involving Rob and Anne results in Rob choosing to Invest and Anne choosing Not Reward, with outcomes of 1 for Rob and 4 for Anne. A Nash equilibrium that is not subgame perfect could involve non-credible threats and suboptimal choices. The change in Anne's payoff leads to an additional subgame perfect equilibrium involving both parties choosing Not Reward.

Step-by-step explanation:

To represent the extensive form of the sequential game involving Rob and Anne using a graphical tree, you would illustrate Rob's initial choice between Invest (I) and Not Invest (NI). Following each of these choices, Anne would then make her decision to either Reward (R) or Not Reward (NR), creating branches for each potential outcome. Rob's and Anne's respective payoffs would be placed at the end of each branch corresponding to the choices made.

In seeking a subgame perfect equilibrium (SPE), we would use backward induction starting from the end of the game. Anne, knowing the payoffs, would choose NR in response to I since 4 > 2 and choose R in response to NI since 2 > 1. Thus, Rob, anticipating Anne's strategies, would choose I since his payoff for I and NR (1) is greater than his payoff for NI and R (0). Therefore, the SPE is for Rob to invest and Anne to not reward. The equilibrium strategies are Rob choosing I and Anne choosing NR, with the outcome being a payoff of 1 for Rob and 4 for Anne.

A Nash equilibrium of this game that is not subgame perfect might involve Anne choosing R after Rob chooses I, even though it is not optimal for her. This would be based on a non-credible threat to reward Rob. However, since Anne gains more by not rewarding, this threat is not credible, making this Nash equilibrium not subgame perfect.

Finally, if Anne's payoff for choosing R after Rob chooses NI changes to 1, the new SPE would involve Anne choosing NR instead of R in response to NI. The new equilibrium strategies would be Rob choosing NI and Anne choosing NR, since now both R and NR give Anne the same payoff but Anne has a higher payoff by choosing NR when Rob invests.