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Chainstore paradox (or "Chain-Store paradox") is a concept that purports to refute standard game theory reasoning.

## The chain store game

A monopolist (Player A) 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):

### Induction theory

Consider 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).

The outcome in the final period is set in stone, so to speak. Now consider period 19, and the potential competitor's decision. He knows that A will cooperate in the next period, regardless of what happens in period 19. Thus, if player 19 enters, an aggressive strategy will be unable to deter player 20 from entering. Player 19 knows this and chooses IN. Player A chooses cooperate.

Of course, this process of backwards 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 a 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 (7×5 + 2×3), which is better than the induction (game theoretically correct) payoff.

If Player A follows the game theory payoff matrix to achieve the optimal payoff, they 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.

### Selten's Response

Reinhard Selten's response to this apparent paradox is to argue that the idea of "deterrence", while irrational by the standards of Game Theory, is in fact an acceptable idea by the rationality that individuals actually employ. Selten argues that individuals can make decisions of three levels: Routine, Imagination, and Reasoning.

### Complete Information?

If we stand by game theory, then the initial description given for the game theory payoff matrix in the chain store game is not in fact the complete payoff matrix. The "deterrence strategy" is a valid strategy for Player A, but it is missing in the initially presented payoff matrix. Game theory is based on the idea that each matrix is modeled with the assumption of complete information: that "every player knows the payoffs and strategies available to other players."

The initially presented payoff matrix is written for one payoff round instead of for all rounds in their entirety. As described in the "deterrence strategy" section (but not in the induction section), Player A's competitors look at Player A's actions in previous game rounds to determine what course of action to take - this information is missing from the payoff matrix. In this case, backwards induction seems like it will fail, because each individual round payoff matrix is dependent on the previous round. In fact, by doubling the size of the payoff matrix on each round (or, quadrupling the amount of choices -- there are two choices and four possibilities per round), we can find the optimal strategy for all players before the first round is played.

## Selten's levels of decision making

### The routine level

The individuals use their past experience of the results of decisions to guide their response to choices in the present. "The underlying criteria of similarity between decision situations are crude and sometimes inadequate". (Selten)

### The imagination level

The individual tries to visualize how the selection of different alternatives may influence the probable course of future events. This level employs the routine level within the procedural decisions. This method is similar to a computer simulation.

### The reasoning level

The individual makes a conscious effort to analyze the situation in a rational way, using both past experience and logical thinking. This mode of decision uses simplified models whose assumptions are products of imagination, and is the only method of reasoning permitted and expected by game theory.

## Decision-making process

### The predecision

One chooses which method (routine, imagination or reasoning) to use for the problem, and this decision itself is made on the routine level.

### The final decision

Depending on which level is selected, the individual begins the decision procedure. The individual then arrives at a (possibly different) decision for each level available (if we have chosen imagination, we would arrive at a routine decision and possible and imagination decision). Selten argues that individuals can always reach a routine decision, but perhaps not the higher levels. Once the individuals have all their levels of decision, they can decide which answer to use...the Final Decision. The final decision is made on the routine level and governs actual behavior.

## The economy of decision effort

Decision effort is a scarce commodity, being both time consuming and mentally taxing. Reasoning is more costly than Imagination, which, in turn is more costly than Routine. The highest level activated is not always the most accurate since the individual may be able to reach a good decision on the routine level, but makes serious computational mistakes on higher levels, especially Reasoning.

Selten finally argues that strategic decisions, like those made by the monopolist in the chainstore paradox, are generally made on the level of Imagination, where deterrence is a reality, due to the complexity of Reasoning, and the great inferiority of Routine (it does not allow the individual to see herself in the other player's position). Since Imagination cannot be used to visualize more than a few stages of an extensive form game (like the Chain-store game) individuals break down games into "the beginning" and "towards the end". Here, deterrence is a reality, since it is reasonable "in the beginning", yet is not convincing "towards the end".

## References

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