Una brigata gioca a palla a 60 el giuoco e 10 p caccia. E fano posta duc 10. Acade p certi acideti che non possano fornire, e l’una pte a 50 e l’altra 20. Se dimanda che tocca p parte de la posta.--Luca Pacioli, Summa de arithmetica, geometria, proportioni et proportionalita, 1494
A group plays a ballgame to 60 and each goal is 10. They stake 10 ducats in all. It happens by certain accidents they are not able to finish; and one party has 50 and the other 20. What portion of the stake is due to each party?
14 comments:
Neither side won, so they each get their own stake back. I.e. 5 ducats to each party, if I understood the setup correctly.
Giving the original stakes back may cause the leading team to complain that their additional goals counted for nothing, and it will give the trailing team an incentive to contrive a way to end the match early as soon as it seems they have little chance of coming back. Can these effects be eliminated with a different distribution of the stake?
Maybe they can be eliminated, but should they be? It's not stated that this is an iterative situation, rather than a one-off, so it can't be assumed that future incentives matter.
Also, the additional goals should have counted for nothing - the payout was for winning with 60 points, not for having a certain differential - so complaining that they count for nothing is bogus.
I say the only fair distribution is 5 ducats each.
Pascal said that "the first thing that we must consider is that the money the players have put into the game no longer belongs to them ... but they have received in return the right to expect that which luck will bring them, according to the rules upon which they agreed at the outset."
Perhaps the trailing side objects to receiving all of their stake back, having so nearly lost. Or perhaps the referee requests a formula so that the incentive to lose will be removed. Is there a way to divide the winnings that is proportional to the standings?
Not saying pascal is wrong, but if they agreed at the outset on how to distribute the stakes in case of the game not being completed, then the question is moot - they simply distribute the ducats according to the agreed-upon scheme.
If they didn't account for that situation then the situation hasn't changed - you still need to come up with the fairest way to distribute the stakes, for some definition of fair.
It could be that the trailing team objects, just as it could be that the leading team might object to not splitting down the middle since they had not, in fact, won. The puzzle as stated really is a question about the reader's intuition regarding fairness.
And I still don't see how the incentive to fail comes in unless either it's an iterative situation (league play, perhaps) or it's specified in advance that a no-result means splitting the stakes evenly - in which case it's specified in advance and the question is moot.
But as to your last question, it depends on whether there are fractional ducats.
Suppose not. Then we can imagine that the payout starts at 5-5, and each goal "pulls" the payout in the direction of the scoring team. E.g. Team A scores so the payout becomes 6 for A and 4 for B. Under this scheme, the payout for this situation would be 8 ducats to team A and 2 to team B.
If there are fractional ducats then we could say that the goal differential is three, so the ducat differential should be three, and split it 6.5-3.5.
Surely there are other schemes, but I don't know that one is more naturally "fair" than any other.
I still say that nobody won so it's a push.
Ideally we will identify a scheme by which the stake is divided when an incomplete match is halted, which can be applied to all future matches and is fair to both sides. This can then be written into the rules in advance of future matches. (Applying the scheme to the current match retroactively may bring complaints.)
Simply returning the original stakes leaves a strong team vulnerable to clever, unsporting teams looking for ways to get a match cancelled when they fall far behind. We want teams to get credit for goals scored, and we can specify a division with arbitrary precision and then round to the nearest ducat.
How do we specify how far each goal "pulls" the payout? One ducat per goal is simple, but doesn't take into account the total number of goals required to win. If they were playing to 100 goals (1000 points), then a game interrupted at a close score of 40 goals to 30 should not result in the entire prize going to the team that was ahead by 10 goals.
We're getting kind of far out in the assumption forest. The original problem said nothing of future matches, games to 1000, etc. The answer to the problem as stated is still a 5-5 split.
Also, I don't think we can dismiss out of hand the option of giving the entire stake to a team that's ahead 40-30 in a game to 100. It would be something like the mercy rule.
But if the problem is reposed as something like "find a general formula for dividing the stakes according to the standings" then maybe this works:
Let the number of total participants be N, let the teams be designated as T1, T2, ... TN, and let the stake wagered by each participant be S1,S2, ..., SN, where T1 wagered S1, T2 wagered S2, ... and TN wagered SN (the stakes may be different because of handicapping, etc.).
Let the number of goals required to win be G
Then let the leading team be designated by TK for some 1 <= K <= N.
Then for each TJ, J != K, let DJ be the number of goals by which team TJ trails team TK.
Then team TK receives all of its stake back, and for each J != K team TJ pays (SJ * DJ)/G to team TK (and team TJ retains the remainder of its stake).
If there are no fractional ducats, round however you like, or carry the fractional parts over to the next match.
Applying this formula to the original problem we get a split of 7.5 to the leading team and 2.5 to the trailing team. Which, of course, is blatantly unfair since the correct split should be 5-5.
Oh, and if two or more teams are tied for the lead they split the proceeds from the other teams in proportion to the stakes they each put up.
I.e. if teams TX, TY, and TZ put up stakes SX, SY, and SZ respectively, and the total contributions from the trailing team(s) stake(s) is W then team TX will get (SX * W)/(SX + SY + SZ), etc.
Perhaps we can forgive "master Luca"* his lack of exactness in specifying what he means by a fair division. Game theory, after all, was hundreds of years away. His discussion (143 kb PDF) advocates an approach similar to yours, but without accounting for the number of goals to be scored.
The only criticisms I could make are that (1) your approach doesn't account for how close to victory the leading team was, with identical distributions for a 10-20 score and a 89-99 score and (2) it's not the "official" solution that Fermat and Pascal worked out. One more piece of information Pacioli should have included is whether the game is one of pure chance or if skill is involved. Fermat and Pascal considered fair coin tosses.
*as Leonardo da Vinci referred to his friend and math tutor Pacioli. My source suggests that Leonardo would struggle in a modern grade school arithmetic class.
Another difficulty with awarding winnings to the leader proportional to the goal differentials is that there could be a significant difference in outcome for teams with an insignificant difference in performance. Suppose there are many poor teams, many mediocre teams, and two good teams that are neck-and-neck throughout the match. When the match is called off early, it happens that Team A is ahead by a slim margin, but moments before Team B was ahead. Because of this accident of timing, Team A collects ducats from all the lesser teams, and Team B collects nothing, indeed has to pay a humiliating trifle to Team A.
Something which Steve quoted Pascal as saying brought to mind an idea for how to completely side-step the issue.
Since the money, once wagered no longer belongs to the combatants (only the winner), why not just place the money in some sort of escrow account until such a time as the contest may be completed?
This may well be giving the game away, but the scheme wished for, if implemented in the long run over many tournaments, will assure both leaders and trailers that the other side has nothing to gain by getting a match to end early. Mathematically.
I think it depends on team's chance of
winning also. But let's say they are equal teams, chance of winning a
game for a team is 50%. In other terms, none of the teams is more
likely to win a match.
Starting from score 5-2 all possible outcomes are {6-2, 6-3, 6-4, 6-5,
5-6}. The chance of second team to win the game is (1/2)^4 = 1/16,
needs to win 4 games consecutively. For the first team it is 15/16. I
think it should be distributed as {10*15/16, 10*1/16} = {9.375,
-0.625}.
If I were to be the decision maker for this game, I would select to
give 5 to each team; from non-mathematic perspective Rasalom is right
imho.
Dividing the stakes based on the expected outcome is the desired approach, as derived by Pascal and Fermat. The story is told briefly at mathforum.org, and probably more accurately on the more staid Wikipedia article.
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