Find the SPNE of the following game

Question

The game works as follows: assume that there are two players, 1 and 2. 1 decides to lie or tell the truth. If 1 tells the truth, 2 needs to decide to take her medications or not. Later, in case 2 did not take her medications or 1 lied to her, 2 needs to decide to go to the hospital or not. If 1 lied to her and she went to the hospital, then the utility of the 1 player is 0 and the second player's utility is 1. If 1 lied to her but she did not go to the hospital, then the first player's utility is 1 and the second player's utility is 0. If 1 told her the truth and she did not take her medications, then her going to hospital generates 1 utility to the first players and 0 utility to the second; not going to the hospital gives both of them 0. If 2 takes her medications after 1 told the truth, then they both get 2. Okay, I used backward induction and this is what I got: if player 1 lies, 2 goes to the hospital. If player 1 tells the truth, player 2 is better off taking the meds. My question is: when stating my SPNE, should I mention that if 2 player decides not to take meds, then she is indifferent between going and not going to the hospital? Because it's clear that the decision to take the meds clearly dominates the decision not to take meds, should I even mention it when stating SPNE?? THANKS FOR YOUR HELP!!

Answer 1

Yes, you should say what each player should do in every subgame of the game. Hagen is essentially analyzing the strategic form of the game, rather than the extensive form.

In the extensive form, working backwards,

• If 1 told the truth and 2 did not take the meds, 2 is indifferent between going to the hospital and not, so going to the hospital with any probability maximizes her payoff, for payoffs of $(p1,0)$ where $p=pr[hospital]$.
  • If 1 lied, 2 strictly prefers the hospital over not, for payoffs of (1,1). If 1 told the truth, taking meds gives 2 a payoff of 2 while not taking the meds gives a payoff of 0, so 2 takes the meds.
  • So if 2 tells the truth, the expected payoff is 1, while if 1 lies, the expected payoff is 1. So 1 strictly prefers to tell the truth.

The SPNE is 1 tells the truth; 2 goes to the hospital if 1 lied and takes the meds if 1 told the truth; if 2 did not take the meds after being told the truth, any randomization over hospital and not is part of an SPNE.

You should listen to your professor. The "stories" that frame games are just stories, you are supposed to learn and apply the equilibrium concepts correctly, not intuitively. The point is to understand what kinds of answers different equilibrium concepts give, and then judge the quality of the answers across many games. If your intuition says there's a problem, then you come up with new equilibrium concepts or refinements to explain why, and that's what game theory "is". But you have to apply the concepts correctly to get the right answer. You might think it is silly in this game, but there are many games where off-path behavior determines the equilibrium, and it really matters.

Answer 2

The strategy of 2 consists of a few "bits":

• When told the truth, will she take her medicine? Or not take her medicine, but go to the hospital? Or neither? Symbolize these by $M, H_T, 0_T$.
• When lied to, will she go to the hospital or not? Symbolize these by $H_L, 0_L$.

In total, 2 has six strategies, and the payouts are as follows: $$\begin{matrix}&0_T0_L&0_TH_L&M0_L&MH_L&H_T0_L&H_TH_L\\ T&(0,0)&(0,0)&(2,2)&(2,2)&(1,0)&(1,0)\\ L&(1,0)&(0,1)&(1,0)&(0,1)&(1,0)&(0,1)\end{matrix} $$

From 2's perspective, $MH_L$ is strictly better than $0_T0_L$ or $H_T0_L$, and at least as good as the other three, hence rationally, 2 will follow strategy $MH_L$. With that in mind, 1 will prefer $T$ over $L$. This will lead to utility $2$ for both players.