Imagine a casino offers a new game called “Guts.” A dealer wants to test the game, so he recruits two strangers (Alice and Bob) for an experiment.

The game works as follows. Alice will secretly write an even integer on a piece of paper, and Bob will secretly write an odd integer. Both are limited to writing numbers less than or equal to 1,000.

They will then simultaneously flip over their papers to reveal their numbers. The person who writes the lower number wins the game and is paid that number of dollars.

The game costs $10 to play and will only be played once. Should Alice and Bob give it a try?

What is the (subgame perfect) Nash equilibrium?

(credit: this game is from Tanya Khovanova’s Math Blog)

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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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It could be a great game

If Alice and Bob could cooperate, the game would be extremely profitable.

If both wrote their maximum values–Alice 1,000 and Bob 999–then Bob would win $999 each time. Bob could then split the winnings with Alice. Both would profit $489.5 after each accounts for the $10 cost to play the game.

But amongst strangers such cooperation cannot be assumed. After Bob is paid, he could just as easily walk away from the table with $989 of profit and thank Alice for being a sucker.

So let us consider the case that neither person can trust the other.

Analyzing the incentives

Let’s say that Alice and Bob informally agree to write their maximum values, but each suspects the other will backstab and walk away with profits.

What is Alice’s best response to Bob writing the number 999?

If Alice writes 1,000, then she loses the game, and the best she can do is get $489.5, so long as Bob honors the deal.

Alice thinks: “What if I undercut Bob by writing 998 and try to win for myself? If Bob keeps to his agreement of writing 999, then I would win get paid $998 for undercutting, and I walk away with a lot more profit than our split of $489.5.”

Alice is not the only one who might worry about the arrangement. Bob could fear Alice might undercut him to 998, so he in turn will pre-empt and consider writing down the number 997.

You can probably deduce the rest of the story by extending the logic through backwards induction. If Alice fears Bob will write 997, then she is best to undercut once more to 996. But Bob could reason this far too, and he’s going to consider writing 995. As each person reasons this process further, they mutually undercut each other and the original agreement erodes.

So we can deduce each person reasons that it’s best to write a smaller and smaller number.

How low will the bidding go??

Finding the equilibrium

We can safely deduce Alice will eliminate large numbers, so she will never write a number larger than 10. Similarly, Bob will never write a number larger than 11.

But what happens then? If Alice writes 10, and Bob writes 11, then Alice ends up with $10 which just covers her cost to play the game. Her net profit is zero. Bob, on the other hand, loses his entire $10 entrance fee.

Bob, therefore, again has a reason to undercut Alice. He can think, “I can either write 11 and lose the game, costing me $10. Or I can instead write 9, which lets me win the game at a net loss of just $1. I know it’s a loss, but I’d rather accept a $1 loss than a $10 loss.”

If Bob is going to write 9, however, then what will Alice choose to do? She would naturally be better off writing the number 8. In that case she wins the game and nets a $2 loss, which is better than the $10 loss if she wrote 10 and let Bob win the game.

By continuing this reasoning, Alice and Bob are further tempted to undercut each other in an attempt to to minimize their losses.

Mercifully there is a limit to the madness. Once Bob gets down to writing 1, and Alice writes 0, the game ends with both parties winning nothing. [Edit 4-10: As Chris points out in the comments, the game probably ends when Alice writes 2 and Bob writes 1. There is no real reason for Alice to win the game with no profit.] At that point, Bob has no incentive to write a negative number (interpreted as him paying more to the casino).

The game theory equilibrium, therefore, is for Alice to write 2, Bob to write 1, and both of them end up winning nothing and each losing $10 for the priviledge of playing the game. This is not a pleasant outcome, but of course, that was the casino’s game all along.