## Tuesday, 31 May 2016

### The toilet seat game

When I blogged my review of William Nicolson's "The Romantic Economist - A Story of Love and Market Forces" last year, I mentioned that I would definitely use the toilet seat game as an example in ECON100 this year. However, the game theory topic is already pretty full, so I didn't manage to squeeze it in. So instead, I thought I would talk about it here.

First, a little bit of background. William tells us about one of his relationships, with Sarah, which hit a bit of a rocky patch with respect to toilet seats, and whether they should be left up or down. You may laugh, but there's been several papers that have explored the game theoretical implications of the toilet seat game (see here and here for two examples).

In the case of Will and Sarah, the decision-making can be thought of as a sequential game, with three decision-making nodes. In the first node of the game, Will decides whether to leave the seat up or down. If he leaves the seat down, the game ends (because Sarah is happy). But if he leaves the seat up, Sarah gets angry. She then has the choice of whether to forgive Will, or punish him by nagging or throwing a minor tantrum. If Sarah forgives Will, then the game ends (and Will is pretty happy, because he got to leave the seat up and avoid the nagging). If instead Sarah punishes Will, then Will has the choice of agreeing to change his behaviour for the future, or ending the relationship. The game as a decision tree (extensive form) is presented below. The payoffs to Will and Sarah are simply ranked from 1 (best) to 4 (worst). Note that in this version of the game, ending the relationship is a worse outcome for Sarah than for Will. [*]

We can solve for the (subgame perfect) Nash equilibrium by using backward induction - essentially we start at the end of the game and work our way backwards, eliminating strategy choices that would not be chosen by each player. In this case, the last play is by Will. He can choose to end the relationship (and receive his 3rd best payoff), or change his behaviour (and receive his worst payoff). So, clearly he would choose to end the relationship. Now, working our way back to Sarah's choice, if she punishes Will then we know that Will is going to end the relationship. So Sarah can choose to forgive Will (and receive her 3rd best payoff), or punish him (which results in Will ending the relationship and Sarah receiving her worst payoff). Given that choice, Sarah will choose to forgive Will. Working our way back to the first node then, Will can choose to leave the seat down (and receive his 2nd best payoff) or leave the seat up (knowing that Sarah will forgive him, and he will receive his best payoff). Given that choice, Will is going to leave the seat up. So, the subgame perfect Nash equilibrium in this case is that Will leaves the seat up, and Sarah forgives him.

Will then goes on to point out that the outcome of this game crucially depends on how Will feels about Sarah. If he really wants to be with Sarah, then the game changes. Now say that ending the relationship is the worst outcome for both Sarah and Will, as showing in the decision tree below.

Solving for equilibrium in this case (again using backward induction), we find that Will would agree to change his behaviour (because that provides him with his 3rd best payoff, compared with the worst payoff if he instead chose to end the relationship). Now, knowing that Will is going to change his behaviour, Sarah would choose to punish him if he leaves the seat up (because she would receive her second best payoff when he agrees to change, which is better than her third best payoff, which she would receive by forgiving Will). Now, knowing that Sarah will punish him, Will decides to leave the seat down (and receive his second best payoff, which is better than the third best payoff that he would receive if he left the seat up, because Sarah would then punish him and he would then change his behaviour).

All in all, this is a great example of a sequential game in action. Of course, the game as presented above is necessarily simplified - in fact this game is a repeated game, so the Nash equilibrium is more complex (as shown in the two papers linked at the beginning). But a great example nonetheless.

*****

[*] Careful readers of Nicolson's book will notice that I have altered the payoffs in these games from those that appear in the book. Specifically, the payoff for Sarah in the case of the Up/Punish/Change outcome is that this is Sarah's second best payoff (whereas in the book it is her third, which is equal to the Up/Forgive outcome). I expect this was a slight error in the book.