Chainstore paradox

Chainstore paradox is a concept which purpose of existence is refusing the standard game theory reasoning. It lies in the debate of a non-credible threat response enter in a rational or irrational model, whether we can use a complete information game or not to model that situation.

In the chain store paradox we have a monopolist (Player A) which has branches in 20 towns. He faces 20 potential competitors, one in each town, who will be able to choose in or out. They do so in sequential order and one at a time. If a potential competitor chooses out, he receives a payoff of 1, while A receives a payoff of 5. If he chooses in, he will receive a payoff of either 2 or 0, depending on the response of Player A to his action. Player A, in response to a choice of in, must choose one of two pricing strategies, cooperative or aggressive. If he chooses cooperative, both player A and the competitor receive a payoff of 2, and if A chooses aggressive, each player receives a payoff of 0.

These outcomes lead to two theories for the game, the induction (game theoretically correct version) and the deterrence theory (weakly dominated theory):

• The induction: the decision to be made by the 20th and final competitor, of whether to choose in or out. He knows that if he chooses in, Player A receives a higher payoff from choosing cooperate than aggressive, and being the last period of the game, there are no longer any future competitors whom Player A needs to intimidate from the market. Knowing this, the 20th competitor enters the market, and Player A will cooperate (receiving a payoff of 2 instead of 0). This process of backward induction holds all the way back to the first competitor. Each potential competitor chooses in, and Player A always cooperates. A receives a payoff of 40 (2×20) and each competitor receives 2.
• Deterrence theory: this theory states that Player A will be able to get payoff of higher than 40. Suppose Player A finds the induction argument convincing. He will decide how many periods at the end to play such a strategy, say 3. In periods 1–17, he will decide to always be aggressive against the choice of IN. If all of the potential competitors know this, it is unlikely potential competitors 1–17 will bother the chain store, thus risking the safe payout of 1 (“A” will not retaliate if they choose “out”). If a few do test the chain store early in the game, and see that they are greeted with the aggressive strategy, the rest of the competitors are likely not to test any further. Assuming all 17 are deterred, Player A receives 91 (17×5 + 2×3). Even if as many as 10 competitors enter and test Player A’s will, Player A will still receive a payoff of 41 (10×0+ 7×5 + 3×2), which is better than the induction (game theoretically correct) payoff.

If Player A follows the game theory payoff matrix to achieve the optimal payoff, he or she will have a lower payoff than with the “deterrence” strategy. This creates an apparent game theory paradox: game theory states that induction strategy should be optimal, but it looks like “deterrence strategy” is optimal instead. The “deterrence strategy” is not a Nash equilibrium: It relies on the non-credible threat of responding to in with aggressive. A rational player will not carry out a non-credible threat, but the paradox is that it nevertheless seems to benefit Player A to carry out the threat.

See also

Game Theory, Selten’s levels of decision making

Papers

• Selten, Reinhard (1978). “The chain store paradox”. Theory and Decision 9 (2): 127-159.
• Ordeshook, Peter C. (1992). “Reputation and the Chain-Store Paradox”. A Political Theory Primer. Routledge. pp. 247-249.