# Game theory (voting game) - why does player 3 have > 4000 strategies?

A committee with three members, {1; 2; 3}, has to choose a new member of a club among a set of four candidates, {a; b; c; d}. Each member of the committee has veto power which is used in a successive way, starting by member 1, and finishing with member 3. Each member of the committee has to veto one and only one of the candidates that have not been vetoed yet.

(a) Draw the extensive form of the game, writing in the terminal nodes the name of the elected candidate.

(b) How many strategies has each player? Do not try to write them out to > count them (player 3 has more than 4,000)

According to the text of part(b) of the problem, player 3 apparently has more than 4000 strategies.

I'm not seeing how that's the case. As I understand it, the game tree should look like this:

So where's this > 4000 strategies coming from? (I'm completely accepting of the fact that I'm probably wrong - just want to understand why!)

• Can Player 3 negotiate before the voting begins? I think you need to provide more context for the " ? $4000$" strategies assertion. The link says you must think about Nash equilibria on preference orderings. Nov 26 '17 at 13:48
• @EthanBolker I've added the full text from parts (a) and (b) of the question. Is it to do with preference orderings then? I'm not seeing where the 4000 part comes from just from the game tree.
– Thev
Nov 26 '17 at 13:53
• @croraf I already solved it! Drop me an email at thevesh.theva@gmail.com and I'll send you the problem and solution!
– Thev
Dec 14 '17 at 7:17

You have the correct game tree. I think the issue is just the definition of "strategy" - a strategy for player 3 is a full description of what that player will do in all possible situations. There are $12$ different situations that player 3 may end up in, corresponding to the $12$ boxes marked "3" in your diagram. For each of these player 3 has $2$ choices, so the total number of different possible strategies is $2^{12}>4000$.