Solution
The Chain Store's Twenty Markets
Show the problem again
A chain store has a branch in each of 20 towns. In turn, in towns 1, 2, ..., 20, a different potential competitor decides whether to enter. Each entrant observes everything that happened in earlier towns. In one town the payoffs (entrant, chain store) are: the entrant stays out: (1, 5); the entrant enters and the chain store accommodates: (2, 2); the entrant enters and the chain store fights: (0, 0). Each town's game is played once, the chain store maximises the sum of its payoffs over all 20 towns, and every entrant maximises his own payoff. Everyone is rational and the game tree is common knowledge. What total payoff does the chain store earn in the subgame-perfect equilibrium?
Worked solution
The answer is 40. In town 20 there is no future, so if the entrant enters, accommodating (2) beats fighting (0). The entrant, foreseeing this, enters (2 > 1). Now town 19: whatever the chain store does there cannot change what happens in town 20, because the town-20 entrant enters regardless. So the chain store again accommodates, and the entrant again enters. Repeating the argument back to town 1, every entrant enters and is accommodated. The chain store earns 2 in each of 20 towns, so 40. The reputation-building strategy of fighting, which would yield 100 if it deterred everyone, is not credible under common knowledge of rationality; this is the chain-store paradox.
Source: Selten's chain-store paradox (1978), in a simplified numerical form. Statement written for AxiomIQ.