Breaking Three-Way Ties

The United States Backgammon Federation (USBGF) has promulgated an interesting method for breaking three-way ties in the tournaments they sanction:

To appreciate what’s going on here, you need to bear in mind that USBGF tourneys generally offer substantial cash prizes. Say, for example, that for a particular tourney the payouts for the top three places are $3000, $2000, and $1000. Assuming that the three players are of equal skill, this means that the the player who gets the bye, A in the example, will win $3000 half the time, and $1000 the other half of the time, with an average expectation of $2000. The other two players will win $3000 a quarter of the time, $1000 another quarter of the time, and $2000 half the time, so that their expectation is also $2000. As long as the prize payout for the second place winner is one third of the total fund for the top three places, the expectations will be equal.

A clever procedure. But is it possible to do better?

The flaw in the tiebreak procedure is that it doesn’t fully equalize the expectation of the players. In addition to prize money, players covet American Backgammon Tour points, which accumulate each year and also over a lifetime. These presumably can be handled the same way, so that’s not a problem.

But one aspect of the tourney cannot be easily equalized: the trophy for the winner, and the right, for every after, to be remembered as the winner of the 2018 Podunk Open Championship. The title is indivisible, and the player who gets the bye will win it twice as often as the other two players.

Let’s say, for example, that the psychic benefit of being the overall winner is worth another $3000 to the players. In that case, the expectation for the player who gets the bye is $3500 – $2000 in prize money, and another $1500 for the half the time they take the title. But the other two players expectation is only $2750, because they get the title a quarter of the time.

Now, if the players could agree on the value of the title, you could adjust the payouts so that the expectations are still equal. Here, you could pay $1500, $3000, and $1500 for first, second, and third, respectively, which figuring in the extra $3000 of psychic benefit would equalize expectations at $3000 apiece. It’s not intuitive, however, to pay first place less than second place.

Furthermore, the psychic benefit of the title is likely to be worth different amounts to different players. Let’s say, for example, that the three players have these preferences:

Player A is a wealthy man, for whom a $3000 prize could be lost as a rounding error in his current account. He plays the game mostly for the prestige of being acknowledged as one of the game’s best players (although he secretly recognizes that he’s not as good as B or C). He knows that three grand is a significant sum to others, and so it’s worth something to him to win it. But, on that scale, the trophy is worth $8000.

Player B, by contrast, is a truly outstanding player, struggling to make a living as a backgammon pro, which tends to be a financially marginal operation. She wants money, the more the better. She may be ambivalent about winning the title – she’s afraid that another big win will scare away some of the richer dilletantes (possibly including A) she relies upon to lose money to her in side games, and thus might assess the psychic value of the title at zero.

Player C is an ordinary strong player, to whom the title is worth $1000.

Now (ignoring, for simplicity sake, the difference in skill between the players) here’s how each player sees their position, assuming that we’re back to the 3-2-1 payout ratio for the money part of the prize:

Player A has an expectation of $4000 if he doesn’t get the bye – the basic $2000, and another $2000 for the quarter of the time that he wins two matches for the title. But if he gets the bye, his expectation rises all the way to $6000.

Player B’s expectation is simply $2000 because the title means nothing to her. Player C gets a small psychic benefit from the title, and so has an expectation of $2500 with the bye, and $2250 without it.

Now, suppose that instead of randomly assigning the bye to one player, we auction it off!

Player A would be willing to pay up to $2000 for the bye. The exact amount depends on how the auction is conducted, but let’s assume that it’s a sealed-bid auction, and further that player A overestimates how much the title is worth to the others, and so bids a healthy $1500 for the bye. Player B bids zero, and player C bids only the $250 benefit he expects.

So, player A wins the bye, and the $1500 he bid is split evenly between B and C. Now, A’s expectation is $4500, B’s is $2750, and C’s is $3000. Everyone is better off than they would be if the bye had gone to either B or C, and the total aggregate value of the expectations is no lower even if A happens to draw the bye!

This is a scheme that an economist would love – by manipulating the rules so as to allow the player who really covets the title to buy a better chance of winning it, everyone does better.

It’s much less clear that this is a practical way to run a tourney. I’m pretty sure I wouldn’t want to have to explain all of this to skeptical players at the tournament.

In any case, it should be noted, that there are (as far as I know) no USBGF tourneys that are capable of producing a three-way tie that don’t also have a well-established procedure for breaking the tie that’s at least as fair as the proposed new system.

So, alas, it appears that the clever method adopted by the USBGF is not so much a solved problem as it is a solution in search of a problem.