# A game of Guts

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)

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.

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• http://www.dangoldin.com Dan Goldin

Reminds me of the auction for a \$20 bill. The winner gets the \$20 after paying the bid bill but the runner up has to pay what he bid. It’s worse than the “Guts” game since the downside isn’t capped.

It’s referred to as the “Dollar Auction” – http://en.wikipedia.org/wiki/Dollar_auction

• Chris

I think the game ends at 2, 1.  There is no point for Alice to write 0.  At \$2 she loses the game and pays \$10.  At \$0 she wins the game and pays \$10.  They have the same payout, and I think that the slight benefits of being generous in letting someone else win outweigh any benefits of “winning” a loser’s game.

• Norcross

From the description it sounds like Alice and Bob have the choice as to whether or not to play – wouldn’t this affect the outcome?  If Bob was going to bid \$1, for example, there is no reason for him to even bother playing – at any bid under \$11, there is no reason to play.  Likewise, if Alice is planning to bid anything less than \$12, she has no reason to play (even \$10 is pointless, since the best she could hope for is to break even while she has a chance to lose \$10).  So if they both choose to play, Bob’s equilibrium bid would be \$11, and Alice’s would be \$12.  The casino would still be happy with this result, since it would get a net profit of \$9.
Being forced to play the game would lead to the worse outcome, but I don’t think the casino would like that anyway since it would pretty much guarantee neither player ever coming to that casino again!

• Sauron

If your assumption going into the game is that you inherently can’t trust your opponent it would make sense (of some form) for Alice to write 0 just to punish Bob for being untrustable.  The better argument is that, in the case that Bob writes 1, Alice gets a payout of 0 for writing either 2 or 0 but if Bob writes any number larger than 2 she can make either \$2 or \$0 and \$2 is clearly the larger number.

• Norcross

Since the game isn’t repeated, there is no logical incentive to punish Bob – especially since, if he turns out to be nice, you’ve just guaranteed neither of you win anyway, making you the untrustworthy jerk.

Your second argument is great, however.  There really is no reason for Alice to bid less than \$2 – the worst case scenario is exactly the same as bidding \$0 and best case is winning at least something.  From a purely logical standpoint It shouldn’t matter to her how much Bob might win.

• Sauron

You can’t fairly say that there’s no logical incentive to punish Bob.  For example, she might value some notion of “justice” and thus derive value from punishing him.  In the case where she is certain he will write \$1 then the only distinguishing result between writing \$2 and \$0 is that she receives that value.  Similarly, if she values that one of the two players receives some money, so as to not just be paying out the house \$20 for no good reason, then she would write \$2 to ensure Bob walks away with the \$1.

You can’t say that these actions are illogical given some pre-determined preferences.  Normally we ignore such preferences when discussing games since they muck up analysis and a given set of preferences applied to a game technically creates a new game that can be analysed.  However, in a case like this, where Alice essentially gets two choices with identical monetary payouts to herself, I don’t think it’s unfair to consider distinguishing preferences such as that.

However, like noted before, unless she can guarantee with probability 1 that Bob will write down \$1 then there *is* a distinguishing factor between \$0 and \$2 and so she should write \$2.

• Balaji

This is the remarkable thing about poker as well easily 4 or 5 players can easily collude against the other 6th player who apparently has bought a huge buy in a Texas Hold em game by secretly indicating the cards they fold each time so that their friend who is still “in” the pot will get a more accurate mathematical probability of the cards he is holding and probably make a better informed decision against the “sucker” player.This type of collusion is almost impossible to identify and even more difficult to prove.