Zero Sum Games
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I don't know about this market and I haven't seen papers
on the subject. Nevertheless I think that the interaction of companies
in any oligopolistic market has some prisoner's dilemma flavor. To makes
things simple think that there are only two credit card companies, V and
M, and that they have only two possible options: they can either compete,
C, or be friendly to the other, F.
Imaging C as making a lot of advertisement, charge
low fees, etc. with the objective of attracting customers from the other
If both companies play F, each gets $4 billion.
If one plays C and the other F, the first one gets a lot of customers,
enough to pay the extra costs, and earns $5 billion, while the second one
gets only $1billion. If both play C, each gets $2billion. See the following
representation of the game, where the first number is the payoff of M and
the second one the payoff of V:
See that whatever V is playing, is better for M to
play C. But this is also true for V, then the solution of this game is
(C, C). Then in equilibrium both companies play C and end with a lower
profit that if they had been friendly.
Please tell me what kind of game you think the plastic payment card industry (say Visa, Mastercard, and Amex) is playing. Many think it is prisoner's dilemma where everyone can do better by cooperating.
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Since both have incentives to try to steal customers
from the other one, they end with the same amount of customers but with
high costs and, then, lower profits.
But this analysis works if the companies only play
once, what is clearly not true in the credit cards market. If companies
play against each other repeatedly and the interest rate is low enough
they can support (F,F) in equilibrium. How can they do that? They promise
each other to play F for ever unless someone plays C. If that happens they
play C for ever.
For more on repeated games see Fudenberg and Tirol's
book on game theory. For seminal papers on oligopolistic repeated games
see Stigler (64), Porter (82) and Green and Porter (84).