Why might the Democrats have held less voting power when the Republicans were divided by the Tea Party? Why might two children suddenly get along just when a parent is about to intervene?

It’s game theory! Voting power works in mysterious ways. Very often the people that hold power end up at a strategic disadvantage. How is that possible?

I will explain the logic carefully in a silly story where 3 people are voting on which pizza topping to order. A game theory analysis illustrates that a person who has the voting power to break ties ends up in a worse outcome than if the outcome was decided randomly.

The model comes from the Chairman Voting game in Myerson’s classic text Game Theory. (Check out my full list of the best game theory books).

The remarkable conclusion is that in some voting games you can end up in a worse outcome when you have extra voting power.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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The Pizza Topping Voting Game

Suppose Alice, Bob, and Charlie are deciding between the pizza choices of pepperoni, sausage, or supreme.

Each has a different preference. Alice prefers pepperoni to sausage to supreme; Bob prefers sausage to supreme to pepperoni; and Charlie prefers supreme to pepperoni to sausage.

Suppose the highest preference gives a utility of 8, the next highest 4, and the lowest 0. Then we can write out the payouts for each person as follows.

For pepperoni, sausage, supreme 8, 4, 0 – Alice

0, 8, 4 – Bob

4, 0, 8 – Charlie

Suppose they vote on which topping to get. What are each person’s average payouts under the following tie-breaking schemes?

Option 1: A topping choice is made at random.

Option 2: Alice gets to break the tie.

Alice has more voting power in option 2. But surprisingly that will make the situation worse for her!

Random Tie-Breaking

Consider the first voting structure. If Alice, Bob, and Charlie vote for their most preferred option, then each topping choice gets one vote. This goes to a tie in which a random choice is made. Each person gets their favorite, next favorite, and least favorite topping choice in equal proportion. This leads to an average payout of 4 = (8 + 4 + 0)/3 for each person.

In fact, each voting for his or her favorite choice is a Nash equilibrium. To prove this, we have to check no person has an incentive to deviate. If any person tries to vote strategically, then that would mean changing a vote from the best option to the second best. This would lead to that person’s second best option getting 2 votes and winning out. The second favorite option has a payout of 4, which is no better than the average payout of 4 from the random topping choice.

Therefore, there is no reason for any person to deviate and each voting for the favorite is a Nash equilibrium. Each topping is selected randomly 1/3 of the time, with an average payout of 4.

Tie-Breaking Power

Now consider the scheme in which Alice gets to break the tie. If each votes for his or her favorite topping, then the resulting vote is a tie. This triggers the tie-breaking procedure where Alice picks her choice of pepperoni. Alice gets 8, Bob gets 0, and Charlie gets 4.

Is this a Nash equilibrium? No! Bob can do better, as we’ll explain, because he will take into account that Alice has the tie-breaking vote and vote differently. We can solve the game by iteratively identifying dominated strategies and deleting them. (For an introduction to this concept, see my post about how my game theory professor gambled $250 to teach a lesson in rationality).

In the first round of reasoning, no one would vote for their least favorite option, as there is no point voting for an option that might yield a 0 payout as anything else would be better. So this means that Alice would never vote for supreme, Bob would never vote for pepperoni, and Charlie would never vote for sausage.

We can delete those strategies and the game is reduced so each person only has two rationalizable voting options.

For pepperoni, sausage, supreme 8, 4, 0 – Alice (never votes supreme)

0, 8, 4 – Bob (never votes pepperoni)

4, 0, 8 – Charlie (never votes sausage)

Now consider Alice’s best move. Bob/Charlie may vote in four possible ways: (sausage/supreme, sausage/pepperoni, supreme/pepperoni, or supreme/supreme). If they both vote for supreme, then supreme will win and Alice’s vote won’t count regardless of what she does. In the other three cases, Alice’s best choice is to vote for pepperoni, which will either win outright, or she’ll win by being the tie-breaking vote. So clearly Alice’s best move is to vote for pepperoni no matter what.

Knowing that Alice will vote for pepperoni, what should Charlie do? Bob will either vote for sausage or supreme. If Bob votes for sausage, then Charlie obviously does not want to vote for sausage and get 0. Charlie can either vote for pepperoni–which wins outright and gets him 4–or Charlie can vote for supreme, and pepperoni will win with Alice’s tie-breaking vote. If Bob votes for supreme, then Charlie would be smart to vote for supreme as well and get his favorite option. So clearly Charlie is smart to vote supreme no matter what.

So what does Bob do? No matter what, Alice votes for pepperoni, and Charlie votes for supreme. If Bob votes for his first choice of sausage, it sends the vote into a tie, in which case the result is Alice will break the tie for pepperoni, which Bob least prefers. So Bob is better off voting for supreme, which means supreme wins and Bob gets 4.

In summary, Alice votes for pepperoni, and Bob and Charlie both vote for supreme. Thus, supreme wins outright and gives payoffs of 0 to Alice, 4 to Bob, and 8 to Charlie.

Paradoxically, this Alice gets 0, which is worse than the average payout of 4 when she did not have the tie-breaking vote! Having extra voting power means Alice always ends up with her least favorite choice.

What just happened?

The logic is straightforward, but the intuition can get lost in the details. Here’s how Alice’s power lead to her own undoing.

Since Alice can break ties, she is confident to vote for her most favored option. This encourages Charlie to always vote for his most favored option–either he will get it, or Alice will break a tied vote to for pepperoni, leading to Charlie’s second favored option.

Bob, who normally votes for his favorite option, does not want the game to go into a tie–if it does Alice will choose pepperoni, his least favored option. So Bob will strategically vote for his second favorite choice of supreme to avoid the tie.

In other words, the fact that Alice can break the tie results in Bob “teaming up” with Charlie to avoid the tie from happening in the first place.

The dynamics of voting lead to the surprising result that Alice is better off relinquishing the tie-breaking vote and letting the tie be decided randomly. You can see how this dynamic could apply to politics–when Congress and the President are from the same party, that could encourage factions from the other party to unite. Or consider parenting–the threat of the parent choosing a really boring activity can inspire kids to resolve differences and play nicely.

So while it seems like more voting power is always a good thing, there are theoretically times when having voting power can instead mobilize the opposition and backfire. In that case, you should do the “noble” thing and explain why it’s fairer to implement a tie-breaking scheme. No one will suspect that you’d give up voting power to get your way. The game theorist acts in mysterious ways.