Ground rules for
game theory:
- All the people in every group act in the best interests of the group
-
No group knows what the other groups will do in advance
-
The groups fully distrust each other
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The groups can sustain considerable short term losses
-
It is known precisely what the payoffs associated with every combination of options are
All types of (perfectly valid) objections can be raised: For example, if you could be in profitable business with somebody you do not trust, there are
contracts. Even an informal agreement with predefined (and perfectly enforceable) consequences on anybody who does not keep their promises.
Early days in Game Theory (some eighty years ago) concentrated on
zero sum games, in which whatever somebody wins, somebody else loses, so there can be no room for negotiation.
The
game matrix is usually associated with a game: In every cell, there is a number associated with the payoff to whoever selects rows, hereafter Anthony, for every combination of options. A negative figure implies a payoff to whoever chooses columns, hereafter Beatrice.
The minimum payoff to Anthony is 0 in row 1 and 2 in row 2, therefore he opts for (2) in row 2. Similarly, the maximum cost to Beatrice is 2 in column 1, and 4 in columns 2 and 3, so she chooses (2) in column 1.
By the
minimax criterion, there is equilibrium at (A2, B1). Players will (individually) move away from it at their loss.
Such a
saddle point need not exist in general, however. Then there is an
optimal mix of options, for example, in the following matrix Anthony plays A1 (38%) and A2 (62%), while Beatrice chooses B1 (62%) and B2 (38%). The optimal mix need not be
unique. Options are taken randomly, but in the proportions indicated.
As the matrices get larger, it is better to get the results from a computer: There is an exact (but elaborate) method and an
iterative one which should converge to a near optimal mix.
Two players both show out one of two cards at the same time, either an ace or a two. If they both show out the same card, Anthony pays 2 units, otherwise Beatrice pays one unit if she shows out the ace and three units if she shows out the two. This must be a fair game- or is it.
The
value of the game is 0.125 to Beatrice.
These pages may be of interest:
I do not know how to swim, I do not get into the sea.
I do not get into the sea, I never learn how to swim. |
For my own code up to this point:
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