The Chairman Paradox

This case analysis could be introduced right after the simultaneous games example. Material is ....

Voting shares with Games the feature of having clearly defined rules and procedures, known to all participants in advance. There are also many participants, the voters, and outcomes. These outcomes are viewed differently by the different voters, therefore there are also payoffs attached to the different possible outcomes, even though these payoffs may not be monetary. Still, what seems to be missing at first sight are the choices. Wouldn't every voter just vote for the option the voter prefers? Actually this is what happens often, and is called "voting honestly".

However, not voting for the most preferred option is not something unheard of. If for example three candidates are running for the mayor position, with the one getting the most votes winning, then you may not vote for your preferred candidate provided this candidate looks chanceless. Then you may vote for your second choice to prevent the third one, which you really dislike. Voting for an option not preferred is called "voting strategically". Since voters can do this---after all, the voters are free, aren't they--- voters have a choice, and voting can always be considered to be a game.

In this chapter we discuss the following very simple voting procedure:

Voting with Chairman

A committee with three members called Ann, Beth, and Cindy has to choose between several options. Every player gives one vote, and the option getting the highest number of votes wins. In case of a tie the chairman, Ann, decides among the tied options.

In this Chapter we assume that there are three options A, B, and C. Ann prefers A over B and B over C. The implicit assumption is that Ann would also prefer A over C then, however in real life this may not be neccesarily so. We also assume that Beth prefers B over C over A, and Cindy prefers C over A over B. We translate this into payoffs by paying 2 units in case the favorite is chosen, 1 unit if the middle preference is selected, and 0 units otherwise. It even turns out that it is not relevant how high these 3 numbers 2, 1, 0, are, you get the same results by using 18, 17, 2, or any other triple of ordered numbers. Using the terminology of Chapter ... , it suffices to have an ordinal scale for the payoffs.

We also assume throughout this Chapter that everybody knows the preferences of everybody. This is called "complete information" in Game Theory. In small groups, like the committee of our example, this may not be unrealistic. Without complete information the analysis would become much more complicated.

If everyone votes honestly her first preference, every option gets exactly one vote, there is a tie, which is broken by chairman Ann by selecting option A. But the voters can vote strategically---not voting for their first preference to achieve a higher payoff. We will analyze how would the three players play, and what option would be selected? (compare [Moulin 1986]).

Main Variant

The first system of these matrices shows the outcome, the elected option, in each case:

Ann votes A
  Cindy votes A Cindy votes B Cindy votes C
Beth votes A  A   A   A 
Beth votes B  A   B   A* 
Beth votes C  A   A*   C 
Ann votes B
  Cindy votes A Cindy votes B Cindy votes C
Beth votes A  A   B   A* 
Beth votes B  B   B   B 
Beth votes C  A*   B   C 
Ann votes C
  Cindy votes A Cindy votes B Cindy votes C
Beth votes A  A   A*   C 
Beth votes B  A*   B   C 
Beth votes C  C   C   C 

"A*" indicates that there has been a tie which was broken by the chairman.

And here is the system of payoff bimatrices:
Ann plays A
  Cindy plays A Cindy plays B Cindy plays C
Beth plays A (2,0,1) (2,0,1) (2,0,1)
Beth plays B (2,0,1) (1,2,0) (2,0,1)
Beth plays C (2,0,1) (2,0,1) (0,1,2)
Ann plays B
  Cindy plays A Cindy plays B Cindy plays C
Beth plays A (2,0,1) (1,2,0) (2,0,1)
Beth plays B (1,2,0) (1,2,0) (1,2,0)
Beth plays C (2,0,1) (1,2,0) (0,1,2)
Ann plays C
  Cindy plays A Cindy plays B Cindy plays C
Beth plays A (2,0,1) (2,0,1) (0,1,2)
Beth plays B (2,0,1) (1,2,0) (0,1,2)
Beth plays C (0,1,2) (0,1,2) (0,1,2)

Chairman-has-to-keep-face Variant

Here the chairman cannot afford (since she would loose her face, or since the voting rules don't allow it) to break the tie with a different vote than she did before.

1 plays A
  3 plays A 3 plays B 3 plays C
2 plays A A A A
2 plays B A B A*
2 plays C A A* C
1 plays B
  3 plays A 3 plays B 3 plays C
2 plays A A B B*
2 plays B B B B
2 plays C B* B C
1 plays C
  3 plays A 3 plays B 3 plays C
2 plays A A C* C
2 plays B C* B C
2 plays C C C C

Actually we get two different games, depending on whether the chairman has to stick to his previous vote by breaking the tie, or whether she can break the tie in favor of A no matter how she voted first. The matrices above give the "keeping the chairman's integrity" variant.

Next we display the payoffs for the different situations. This is the description of the game (again the variant where the chairman has to stick to her previous vote in breaking the tie):

1 plays A
  3 plays A 3 plays B 3 plays C
2 plays A (2,0,1) (2,0,1) (2,0,1)
2 plays B (2,0,1) (1,2,0) (2,0,1)
2 plays C (2,0,1) (2,0,1) (0,1,2)
1 plays B
  3 plays A 3 plays B 3 plays C
2 plays A (2,0,1) (1,2,0) (1,2,0)
2 plays B (1,2,0) (1,2,0) (1,2,0)
2 plays C (1,2,0) (1,2,0) (0,1,2)
1 plays C
  3 plays A 3 plays B 3 plays C
2 plays A (2,0,1) (0,1,2) (0,1,2)
2 plays B (0,1,2) (1,2,0) (0,1,2)
2 plays C (0,1,2) (0,1,2) (0,1,2)

IEWD (iterated eliminated of weakly dominated moves) analysis

Let's first analyse whether some moves are strongly or weakly dominated. Actually initially there is no strong dominance. For player 1, move A weakly dominates both move B and move C. Move B also weakly dominates move C. For player 2, only move B weakly dominates move A, otherwise there is no dominance there. For player 3 the only dominance is move C weakly dominating move B.

What happens if we repeatedly eliminate weakly dominated moves? If we start eliminating the weakly dominated moves B and C for player 1, we obtain the following reduced game:

1 plays A
  3 plays A 3 plays B 3 plays C
2 plays A (2,0,1) (2,0,1) (2,0,1)
2 plays B (2,0,1) (1,2,0) (2,0,1)
2 plays C (2,0,1) (2,0,1) (0,1,2)

For player 3, move C dominates both moves A and B, so let's also eliminate them.

1 plays A
  3 plays C
2 plays A (2,0,1)
2 plays B (2,0,1)
2 plays C (0,1,2)

Then player 2 will play move C, and C is elected.

What happens if A plays a little more unpredictable, and is not willing to eliminate weakly dominated moves so quickly. If we start eliminating move A for player 2 and move B for player 3, we obtain the following reduced game:

1 plays A
  3 plays A 3 plays C
2 plays B (2,0,1) (2,0,1)
2 plays C (2,0,1) (0,1,2)
1 plays B
  3 plays A 3 plays C
2 plays B (1,2,0) (1,2,0)
2 plays C (1,2,0) (0,1,2)
1 plays C
  3 plays A 3 plays C
2 plays B (0,1,2) (0,1,2)
2 plays C (0,1,2) (0,1,2)

After that, move A is weakly dominated by move C for player 3. If we eliminate this as well, we get

1 plays A
  3 plays C
2 plays B (2,0,1)
2 plays C (0,1,2)
1 plays B
  3 plays C
2 plays B (1,2,0)
2 plays C (0,1,2)
1 plays C
  3 plays C
2 plays B (0,1,2)
2 plays C (0,1,2)

Now there is still no further dominance for player 2, so if we want to reduce the game further, we have to accept that player 1 now eliminates the weakly dominated moves B and C. After that, player 2 would eliminate (now strongly) dominated move B and we arrive at the same solution: player 1 votes for A and the other two players vote for C, and C wins.

Quite paradoxically C will always win, giving the least payoff to the chairman player 1.

Less is More: Improving by restricting your choices

The chairman opens the session by a little speech she prepared: "Dear friends. last night, when thinking about the election today, it occured to me a little unfair to have the chairman advantage over you. For this reason I decided not to vote for option A today." What will the result of this be?

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