**Repeated games approach.**__Concept.__

As I have already mentioned, in practice firms are likely to interact repeatedly. Such factors as technological know-how, durable investments and entry barriers promote long-run interactions among a relatively stable set of firms, and this is especially true for the industries with only a few firms. With repeated interaction every firm must take into account not only the possible increase in current profits, but also the possibility of a price war and long-run losses when deciding whether to undercut a given price directly or by increasing its output level. Once the instability of collusion has been formulated in the “one-shot” prisoners dilemma game, it raises the question of whether there is any way to play the game in order to ensure a different, and perhaps more realistic, outcome. Firms do in practice sometimes solve the co-ordination problem either via formal or informal agreements. I would focus on the more interesting and complicated case of how collusive outcomes can be sustained by non-co-operative behaviour (informal), i.e. in the absence of explicit, enforceable agreements between firms. We have seen that collusion is not possible in the “one-shot” version of the game and we will now stress upon a question of whether it is possible in a repeated version. The answer depends on at least four factors:

- Whether the game is repeated infinitely or there is some finite number of times;
- Whether there is a full information available to each firm about the objectives of, and opportunities available to, other firms;
- How much weight the firms attach to the future in their calculations;
- Whether the “cheating” can/can not be detected due to the knowledge/lack of knowledge about the prior moves of the firms rivals.

The fact of repetition broadens the strategies available to the players,

because they can make their strategy in any currant round contingent on the others play in previous rounds. This introduction of time dimension permits strategies, which are damaging to be punished in future rounds of the game. This also permits players to choose particular strategies with the explicit purpose of establishing a reputation, e.g. by continuing to co- operate with the other player even when short-term self-interest indicates that an agreement to do so should be breached.

__b.) Finite game case.__

But repetition itself does not necessarily resolve the prisoners dilemma. Suppose that the game is repeated a finite number of times, and that there is complete and perfect information. Again, we assume firms to maximise the (possibly discounted) sum of their profits in the game as a whole. The collusive low output for the firms again, unfortunately for the firms, could not be sustained. Suppose, they play a game for a total of five times. The repetition for a predetermined finite number of plays does nothing to help them in achieving a collusive outcome. This happens because, though each player actually plays forward in sequence from the first to the last round of the game, that player needs to consider the implications of each round up to and including the last, before making its first move. While choosing its strategy its sensible for every firm to start by taking the final round into consideration and then work backwards. As we realise the backward induction, it becomes evident that the fifth and the final round of the game would be absolutely identical to a “one-shot” game and, thus, would lead to exactly the same outcome. Both firms would cheat on the agreement at the final round. But at the start of the fourth round, each firm would find it profitable to cheat in this round as well. It would gain nothing from establishing a reputation for not cheating if it knew that both it and its rival were bound to cheat next time. And this crucial fact of inevitable cheating in the final round undermines any alternative strategy, e.g. building a reputation for not cheating as the basis for establishing the collusion. Thus cheating remains the dominant strategy.

** NOTE:* the *is* however one assumption about slightly incomplete information, which allows collusive outcome to *occur** *in the finitely repeated game, but I will left it for the discussion some paragraphs later.

__c.)_ Infinite game case.__

Now lets consider the infinitely repeated version of the game. In this kind of game there is always a next time in which a rivals behaviour can be influenced by what happens this time. In such a game, solutions to the problems represented by the prisoners dilemma are feasible.

*i.) “Trigger” strategy*

Suppose that firms discount the future at some rate “w”, where “w” is a number between O and 1. That is, players attach weight “w” to what happens next period. Provided that “w” is not too small, it is now possible for non-co-operative collusion to occur. Suppose that firm B plays “trigger” strategy, which is to choose low output in period 1 and in any subsequent period provided that firm A has never produced high output, but to produce high output forever more once firm A ever produces high output. That is B co-operates with A unless A “defects”, in which case B is triggered into perpetual non-co-operation. If A were also to adopt the “trigger” strategy, then there would always be collusion and each firm would produce low output. Thus the discounted value of this profit flow is:

2+2w+2w^2+2w^3+…=2/(1-w)

If fact A gets this pay-off with any strategy in which he is not the first to defect. If A chooses a strategy in which he defects at any stage, then he gets a pay-off of 3 in the first period of defection (as B still produces low output), and a pay-off of no more than 1 in every subsequent period, due to B being triggered into perpetual non-co-operation. Thus, As pay-off is at most

3+w+w^2+w^3+…=3+w/(1-w)

If we will compare these two results, we will get that it is better not to defect so long as

W > (or =) ½

We can conclude that is the firms give enough weight to the future, then non-co-operative collusion can be sustained, for example, by “trigger” strategies. The “trigger” strategies constitute a Nash equilibrium = self-sufficient agreement. However it is not enough for a firm to announce a punishment strategy in order to influence the behaviour of rivals. The strategy that is announced must also be credible in the sense that it must be understood to be in the firms self-interest to carry out its threat at the time when it becomes necessary. It must also be severe in a sense that the gain from defection should be less than the losses from punishment. But because it is possible that mistakes will be made in detecting cheating (if, for example, the effects of unexpected shifts in output demand are misinterpreted as the result of cheating), the severity of punishment should be kept to the minimum required to deter the act of cheating.

*ii.) Tit-for-Tat.*

Trigger strategies are not the only way to reach the non-co-operative collusion. Another famous strategy is Tit-for-Tat, according to which a player chooses in the current period what the other player chose in the previous period. Cheating by either firm in the previous round is therefore immediately punished by cheating, by the other, in this round. Cheating is never allowed to go unpunished. Tit-for-Tat satisfies a number of criteria for successful punishment strategies. It carries a clear threat to both parties, because it is one of the simplest conceivable punishment strategies and is therefore easy to understand. It also has the characteristics that the mode of punishment it implies does not itself threaten to undermine the cartel agreement. This is because firms only cheat in reaction to cheating be others; they never initiate a cycle of cheating themselves. Although it is a tough strategy, it also offers speedy forgiveness for cheating, because once punishment has been administered the punishing firm is willing once again to restore co-operation. Its weakness is in the fact that information is imperfect in reality, so it is hard to detect whether a particular outcome is the consequence of unexpected external events such as a lower demand than forecast, or cheating, Tit-for-Tat has a capacity to set up a chain reaction in a response to an initial mistake.

__d.) Finite game case, Kreps approach.__

Lets now return to the question of how collusion might occur non-co-operatively even in the finitely repeated game case. Intuition said that collusion could happen- at least at the earlier rounds- but the game theory apparently said that it could not. Kreps et al. (1982) offered the elegant solution to this paradox. They relax the assumption of complete information and instead suppose that one player has a small amount of doubt in his mind as to the motivation of the other player. Suppose A attaches some tiny probability p to B referring- or being committed- to playing the “trigger” strategy. In fact it turns out that even if p is