# Determining subgame perfect Nash equilibriums

Question

Three houses share exclusive access to a beach, but it is dirty due to trash washed ashore. A beach clean-up exercise costs $$100$$, but has a value of $$200$$ to each household. A clean-up company offers to take on the exercise and suggests that contributions be made sequentially. First, Household 1 will contribute some amount that is $$x_1$$. Then, after observing $$x_1$$, Household 2 will contribute some amount that is $$x_2$$. Finally, after observing $$x_1$$ and $$x_2$$, Household 3 will contribute some amount that is $$x_3$$. If $$x_1 + x_2 + x_3 \geq 100$$, then the company will go ahead with the clean-up and keep any proceeds. If $$x_1 + x_2 + x_3 \leq 100$$, then the company keeps all contributions and the clean-up is not done.

Find the subgame perfect Nash equilibrium.

Consider Household 1. Observe that it is always in Household 1's best interest to have the beach cleaned, since $$200 > 100$$, so he should offer $$100$$. Now, Household 2 sees this and knows that enough contribution has been made for the clean-up to happen, since $$100 \geq 100$$, so he will offer $$0$$. A similar argument can be made for Household 3. Thus, the equilibrium outcome is $$\{x_1 = 100, x_2 = 0, x_3 = 0\}$$.

Note

I know that the question asked for the subgame perfect Nash equilibrium, but my professor has specifically stated that, for the purposes of the module we are taking, being able to come up with the equilibrium outcome is sufficient (i.e. We do not know how to solve for the actual subgame perfect Nash equilibrium).

I have two questions.

1. Is my equilibrium outcome correct?
2. May I know if my reasoning is sufficient/complete/logical to arrive at the outcome I had reached?

We have just covered game theory, so I am still trying to get used to answering such questions. Any help/thoughts on my answer will be greatly appreciated :)

• For checking whether your solution is a Nash equilibrium is not all you need to check is whether any player benefits individually from changing her strategy (action) given others' strategies (actions). You will find that your solution is indeed an equilibrium. Let me also suggest starting with household 3 with similar reasoning. – Dayne Nov 2 '20 at 16:25
• Also please add a self study tag as it appears to be homework. – Dayne Nov 2 '20 at 16:25
• @Dayne But on further thought, for my solution, wouldn’t Household 1 choose to deviate? Since he knows any amount he contributes less by would be topped up by either Households 2 and/or 3. So I would say the actual SPNE outcome is $\{x_1 = 0, x_2 = 0, x_3 = 100 \}$. – Ethan Mark Nov 3 '20 at 2:37
• That would be SPNE. For NE consideration, we think of this as: would the player deviate given other players' actions. From that perspective all $x_1, x_2, x_3$ such that $x_1+x_2+x_3=100$ are NE. But not SPNE of course. – Dayne Nov 3 '20 at 2:42
• @Dayne I see. So just to clarify, both $\{x_1 = 100, x_2 = 0, x_3 = 0\}$ and $\{x_1 = 0, x_2 = 0, x_3 = 100\}$ are NE but $\{x_1 = 100, x_2 = 0, x_3 = 0\}$ is the only SPNE? – Ethan Mark Nov 3 '20 at 2:48

Just for sake of acknowledgement, please note that the game described in the question is a variation of the famous Ultimatum game. Knowing this can help you get a ton of literature on such games.

Further note that your professor has made an extremely important point that coming up with answer is sufficient, solving is not necessary. My answer is also limited to showing that a given action profile is equilibrium (whether NE or SPNE). Solving games (such as these) is different ballgame altogether (something in which I have no expertise).

For Nash Equilibrium:

To check whether a given action profile is an NE or not, it suffices to show that each player's response is the best response (BR) given other players' actions. Now consider the set of action profiles:

$$X := \{(x_1,x_2,x_3) \,\,|\,\, x_1 + x_2+x_3 = 100\}$$

For any $$x \in X$$, we can see that pay-off for each player is $$200$$ and no player can do any better by changing their action, given other players' action. Hence all action in $$X$$ are NE (note here that since this game is sequential and not simultaneous, we are not considering mixed strategy profiles).

The problem is that intuitively, this does not seem reasonable to us because for player 1, offering anything more than $$0$$ doesn't seem smart.

This is where the refinement of NE, Subgame Perfect Nash Equilibrium comes in:

In SPNE, the equilibrium should be an NE for each subgame of the game as well. This puts some restrictions and is thus a smaller set. In the above game, since player 3's best response is to play $$100-x_1-x_2$$ (for example, if player 3 insists that she'll play $$0$$ if others don't pay $$33.33$$ each, it is really a non-credible threat because it would be irrational for her to play like that), the best response of player 2 and player 1 becomes, $$0$$ for each.

Therefore, the only subgame perfect NE is $$(0,0,100)$$