Imagine a strategic situation in which three legislators vote sequentially on whether they should receive a pay raise. Let's assume that decisions are made by majority rule. If at least two legislators vote yes, then each legislator will receive a pay raise. Although all the legislators would like to receive a pay raise, each knows that they will pay a cost with their constituents if they are seen to vote for the raise. From the perspective of the legislators, four possible outcomes can occur. The most preferred outcome for the legislators is that they get the pay raise even though they personally vote no. The worst possible outcome is that they do not get the pay raise and they vote yes. Let us assume that of the two remaining outcomes, the legislators prefer the outcome in which they get the pay raise when they voted yes to the outcome in which they do not get the pay raise when they vote no. As a result, the preference ordering for each legislator is:
$\text{Get raise, vote no}$ $>$ $\text{Get raise, vote yes}$ $>$ $\text{No raise, vote no}$ $>$ $\text{No raise, vote yes}$
Imagine that you are one of the legislators. Would you prefer to vote first, second or third?
My attempt:
I have assigned the following ordinal payoffs to the four possible outcomes:
$\text{(Get raise, vote no)} = 4$
$\text{(Get raise, vote yes)} = 3$
$\text{(No raise, vote no)} = 2$
$\text{(No raise, vote yes)} = 1$
Then, I have drawn the following game tree:
Here, Legislator 1's payoff is shown first, Legislator 2's payoff is shown second and Legislator 3's payoff is shown third. The choice a Legislator should make in each sub-game is shown in bold.
Obviously, the expected outcome of the game is that Legislator 1 votes no, Legislator 2 votes yes and Legislator 3 votes yes $\text{(no, yes, yes)}$.
Given all of this, how should I decide which legislator I would like to be?
You want to be the legislator with the most preferred outcome. That's the legislator who gets the raise while still getting to vote no, which is the first legislator.