[SumZero]

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For all you game theorists out there.
Here is a simple game with two players, A and B. A is given $1000 and is able
to split it between he and B any way he chooses. B has the option to accept or
reject the offer. If he accepts the offer, the money is divided according to
A's split. If B rejects the offer, both players get nothing.
At face value, this game isn't interesting if played one time. B should always
accept. However, if this game is played 1000 times. it gets more interesting. Now consider that there will be 10 games going on at the same time with 10 different As and 10 different Bs. For the competition, the As are competing against the other As, not their actual opponent in the game.
What is A's optimal offer strategy? What is B's optimal strategy?

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I'm not sure the actual team competition matters to the game or at least you have not formalized how it matters. I am going to proceed by assuming that all As and Bs care about is their playoffs.
To solve these types of games it makes sense to start at the back and work forward.
What would happen in the 1000th period? A would offer B a the smallest possible share and B would accept.
What about the 999th period? Both players know that A will offer B the smallest share in the 1000th period and that B will accept. Knowing this, there is no reason for A to offer B anything but the smallest share in the 999th period.
What about the 998th period? Same thing - both players know A will offer B the smallest share in the last two periods and the B will accept - so in the 998th period A offers B the smallest share and B accepts.
Actually I think its the same thing for all prior periods.
When do you get someting different? If the game was played for an infinite (or indefinite) number of periods you would get different equillibrium strategies.
Passion

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Well this is a case where your theoretical optimal strategy under the assumption that both players are super rational actors is potentially flawed. If you run this game as an experiment many B's will reject offers that they don't feel are fair. This seems may seem irrational to people who only value the units and don't realize that people are not rational actors and that money isn't always a "good" or isn't always the only "good". Some people will turn the money down on a matter of principle.

Also, even if people are relatively rational, B may be able to tell A and convince him that he'll reject any offer that is less than X% of the prize pool. If B follows through on it the first few times rejecting the offer will A continue to offer a 999-1 split? Should he? If B knows that A will change his behavior then B should have this spite strategy. But if A counters and says he'll always offer 999-1 no matter what and if B trusts this and if all B cares about is maximizing his own money and no outside values (like spite or fairness or whatnot) than the 999-1 split is the theoretical optimal. So again this is a case of theoretical optimal versus experimental optimal.

Our economics class ran this game (as a game, with no real money just play money) and iirc tended to find that nearly everyone would accept a 50-50 split, most people would accept a 60-40 split, about half would accept a 2/3-1/3 split, and about a third would accept any split where they got any amount of money. So with that population an offer of somewhere around 60-40 was best.