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## November 12, 2004

### Voting Cycles: A Game Theory Problem

I've just been inspired, on reading a draft chapter of Burt Monroe's *Electoral Systems in Theory and Practice, * to write up a long game theory problem for the next edition of Games and Information. It gets very technical, but I'll post it in case anybody might be interested.

Uno, Duo, and Tres are three people voting on whether the budget devoted to a project should be Increased, kept the Same, or Reduced. Their payoffs from the different outcomes, given below, are not monotonic in budget size. Uno thinks the project could be very profitable if its budget were increased, but will fail otherwise. Duo mildly wants a smaller budget. Tres likes the budget as it is now.

Uno Duo Tres

Increase 100 2 4

Same 3 6 9

Reduce 9 8 1

Each of the three voters writes down his first choice. If a policy gets a majority of the votes, it wins. Otherwise, *Same* is the chosen policy.

(a) Show that (*Same, Same, Same*) is a Nash equilibrium. Why does this equilibrium seem unreasonable to us?

... I continue to have severe Movable Type problems. I can't do Extended entries, so for more, go to Problem 4.7 in this page. When I've got time, I'll think about whether to switch weblog software.

November 14: Now maybe this will work:

ANSWER. The policy outcome is *Same* regardless of any one player's deviation. Thus, all three players are indifferent about their vote. This seems strange, though, because Uno is voting for his least-preferred alternative. Parts (c) and (d) formalize why this is implausible.

(b) Show that (*Increase, Same, Same*) is a Nash equilibrium.

ANSWER. The policy outcome is *Same*, but now only by a bare majority. If Uno deviates, his payoff remains 3, since he is not decisive. If Duo deviates to *Increase*,*Increase* wins and he reduces his payoff from 6 to 2; if Duo deviates to *Reduce*, each policy gets one vote and *Same* wins because of the tie, so his payoff remains 6. If Tres deviates to Increase, *Increase* wins and he reduces his payoff from 9 to 4; if Tres deviates to *Reduce*, each policy gets one vote and *Same* wins because of the tie, so his payoff remains 9.

(c) Show that if each player has an independent small probability epsilon of ``trembling'' and choosing each possible wrong action by mistake, (*Same, Same, Same*) and (*Increase, Same, Same*) are no longer equilibria.

ANSWER. Now there is positive probability that each player's vote is decisive. As a result, Uno deviates to *Increase*. Suppose Uno himself does not tremble. With probability epsilon (1-epsilon) Duo mistakenly chooses *Increase* while Tres chooses *Same*, in which case Uno's choice of *Increase* is decisive for *Increase* winning and will raise his payoff from 3 to 100. With the same probability, epsilon (1-epsilon), Tres mistakenly chooses *Increase* while Duo chooses *Same*. Again, Uno's choice of *Increase* is decisive for *Increase* winning. Thus, (*Same, Same, Sam*e) is no longer an equilibrium.

(With probability epsilon*epsilon, both Duo and Tres tremble and choose *Increase* by mistake. In that case, Uno's vote is not decisive; *Increase* wins even without his vote.)

How about (*INCREASE, SAME, SAME*)? First, note that a player cannot benefit by deviating to his least-preferred policy.

Could Uno benefit by deviating to *Reduce*, his second-preferred policy? No, because the probability of trembles that would make his vote for *Reduce* decisive is 2*epsilon (1-epsilon), as in the previous paragraph, and he would rather be decisive for *Increase* than for *Reduce*.

Could Duo benefit by deviating to *Reduce*, his most-preferred policy? If no other player trembles, that deviation would leave his payoff unchanged. If, however, one of the two other players trembles to *Reduce* and the other does not, which has probability 2*epsilon (1-epsilon). then Duo's voting for *Reduce* would be decisive and *Reduce* would win, raising Duo's payoff from 6 to 8. Thus, (*Increase, Same, Same*) is no longer an equilibrium.

Just for completeness, think about Tres's possible deviations. He has no reason to deviate from *Same*, since that is his most preferred policy. *Reduce* is his least-preferred policy, and if he deviates to *Increase*, *Increase* will win, in the absence of a tremble, and his payoff will fall from 9 to 4-- and since trembles have low probability, this reduction dominates any possibilities resulting from trembles.

(d) Show that (*Reduce, Reduce, Same*) is a Nash equilibrium that survives each player has an independent small probability epsilon of ``trembling'' and choosing each possible wrong action by mistake.

ANSWER. If Uno deviates to *Increase* or *Same*, the outcome will be *Same* and his payoff will fall from 9 to 3 If Duo deviates to *Increase* or *Same*, the outcome will be *Same* and his payoff will fall from 8 to 6. Tres's vote is not decisive, so his payoff will not change if he deviates. Thus, (*Reduce, Reduce, Same*) is a Nash equilibrium

How about trembles? The votes of both Uno and Duo are decisive in equilibrium, so if there are no trembles, they lose by deviating, and the probability of trembles is too small to make up for that. Tres is not decisive unless there is tremble. With probability 2*epsilon (1-epsilon) just one of the other players trembles and chooses *Same*, in which case Duo's vote for *Same* would be decisive; with probability 2*epsilon (1-epsilon) just one of the other players trembles and chooses *Increase*, in which case Duo's vote for *Increase* would be decisive. Since Tres's payoff from *Same* is bigger than his payoff from *Increase*, he will choose *Same* in the hopes of a tremble.

(e) Part (d) showed that if Uno and Duo are expected to choose *Reduce*, then Tres would choose *Same* if he could hope they might tremble-- not *Increase*. Suppose, instead, that Tres votes first, and publicly. Construct a subgame perfect equilibrium in which Tres chooses *Increase*. You need not worry about trembles now.

ANSWER. Tres's strategy is just an action, but Uno and Duo's strategies are actions conditional upon Tres's observed choice.

Tres: *Increase*

Uno: *Increase|Increase;Reduce|Same, Reduce|Reduce *

Duo: *Reduce|Increase; Reduce|Same, Reduce|Reduce*

Uno's equilibrium payoff is 100. If he deviated to *Same|Increase* and Tres chose *Increase*, his payoff would fall to 3; if he deviates to *Reduce|Increase* and Tres chose *Increase*, his payoff would fall to 9. Out of equilibrium, if Tres chose *Same*, Uno's payoff if he responds with *Reduce* is 9, but if he responds with *Same* it is 4. Out of equilibrium, if Tres chose *Reduce*, Uno's payoff is 9 regardless of his vote.

Duo's equilibrium payoff is 2. If Tres chooses *Increase*, Uno will choose *Increase* too and Duo's vote does not affect the outcome. If Tres chooses anything else, Uno will choose *Reduce* and Duo can achieve his most preferred outcome by choosing *Reduce*.

(f) Consider the following voting procedure. First, the three voters vote between *Increase* and *Same*. In the second round, they vote between the winning policy and *Reduce*. If, at that point, *Increase* is not the winning policy, the third vote is between *Increase* and whatever policy won in the second round.

What will happen? (watch out for the trick in this question!)

ANSWER. If the players are myopic, not looking ahead to future rounds, this is an illustration of the Condorcet paradox. In the first round, *Same* will beat *Increase*. In the second round, *Reduce* will beat *Same*. In the third round, *Increase* will be *Reduce*. The paradox is that the votes have cycled, and if we kept on holding votes, the process would never end.

The trick is that this procedure does * not* keep on going-- it only lasts three rounds. If the players look ahead, they will see that *Increase* will win if they behave myopically. That is fine with Uno, but Duo and Tres will look for a way out. They would both prefer *Same* to win. If the last round puts *Same* to a vote against *Increase*, *Same* will win. Thus, both Duo and Tres want *Same* to win the second round. In particular, Duo will {\it not} vote for *Reduce* in the second round, because he knows it would lose in the third round.

Rather, in the first round Duo and Tres will vote for *Same* against *Increase*; in the second round they will vote for *Same* against *Reduce*; and in the third round they will vote for *Same* against *Increase* again.

This is an example of how particular procedures make voting deterministic even if voting would cycle endlessly otherwise. It is a little bit like the T-period repeated game versus the infinitely repeated one; having a last round pins things down and lets the players find their optimal strategies by backwards induction.

Arrow's Impossibility Theorem says that social choice functions cannot be found that always reflect individual preferences and satisfy various other axioms. The axiom that fails in this example is that the procedure treat all policies symmetrically-- our voting procedure here prescribes a particular order for voting, and the outcome would be different under other orderings.

(g) Speculate about what would happen if the payoffs are in terms of dollar willingness to pay by each player and the players could make binding agreements to buy and sell votes. What, if anything, can you say about which policy would win, and what votes would be bought at what price?

ANSWER. Uno is willing to pay a lot more than the other two players to achieve his preferred outcome, He would be willing, to deviate from any equilibrium in which *Increase* would lose by offering to pay 20 for Duo's vote. Thus, we know *Increase* will win.

But Uno will not have to pay that much to get the vote. We have just shown that *Increase* will win. The only question is whether it is Duo or Tres that has his payoff increased by a vote payment from Uno. Duo and Tres are thus in a bidding war to sell their vote. Competition will drive the price down to zero! See Ramseyer & Rasmusen (1994).

This voting procedure, with vote purchases, also violates one of Arrow's Impossibility axioms-- his ``Independence of Irrelevant Alternatives'' rules out procedures that, like this one, rely on intensity of preferences.

Posted by erasmuse at November 12, 2004 03:06 AM

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