Топик: Topic: Is Collusion Possible?

Although the importance of the other players’ choices takes place, sometimes a player has a strategy that is the best irrespective of what others do. This strategy is called dominant, and the other inferior ones are called dominated. A situation in which each player is choosing the best strategy available to him, given the strategies chosen by others, is called a Nash equilibrium. This equilibrium corresponds to the idea of self-fulfilled expectations, tacit, self-supporting agreement. If the players have somehow reached Nash equilibrium, then none would have an incentive to depart from this agreement. Any agreement that is not a Nash equilibrium would require some enforcement.

b.) The problem of collusion.

Now I would like to use an example of a game in which the Cournot output deciding duopoly is involved. This game is illustrated by the table below:

Firm B’s output level
HIGH	LOW
Firm A’s output level	HIGH	(1;1)	(3;0)
	LOW	(0;3)	(2;2)

Here a firm chooses between two alternatives: high and low output strategies. The corresponding pay-offs (profits) are given in the boxes. In this game, the best thing that can happen for a firm is to produce a high level of output while its rival produces low. Low output of the rival provides that price is not driven down too much, thus a firm could earn a good profit margin. The worst thing for a firm is to change places with its rival assuming the same situation takes place. If both firms produce high levels of output, then the price would be low, allowing each of them to earn still positive but very small profits. Nevertheless, (HIGH;HIGH) would be the dominant strategy of this game (we would observe a Nash equilibrium in strictly dominant strategies here). It is the best response of firm A whenever B produces a high or low output and this is also true for firm B. The non-co-operative outcome for each firm would be to get the pay-off of 1. But as we can see, it would be better for both to lower their output and thereby to raise price, as their profits would increase to 2 for each firm instead of 1 in NE. Strategy (LOW;LOW) would be the collusive outcome. The problem of collusion is for the firms to achieve this superior outcome notwithstanding the seemingly compelling argument that high output levels will be chosen.

This was an example of a “one-shot” game and we saw that the collusive outcome was not available for that case. But in reality these games are being played over and over (on a long-term basis) and we will see later in this essay how the collusion can be sustained by threats of retaliation against non-co-operative behaviour.

c.) Predatory pricing.

Here we need to introduce the explicit order of moves in the model. There are again two players-firms on the market- an incumbent firm and a potential entrant in the market. The game is illustrated below:

The potential entrant chooses between entering and staying out of the industry. In the case of his entering, the incumbent firm can either fight this entry (which as we see would be costly to both), or acquiesce and arrive at some peaceful co-existence (which is obviously more profitable). The best thing for incumbent is for entry not to take place at all. There are in fact two Nash equilibria: (IN;ACQUIESCE) and (OUT;FIGHT). But the last mentioned (OUT;FIGHT) is implausible, as if the incumbent were faced with the fact of entry, it would more profitable for him to acquiesce rather than to fight the entry. Due to this fact the potential entrant would choose to enter the industry and the only equilibrium would be (IN;ACQUIESCE). Thus we can conclude, that in this case the incumbent’s threat to fight was empty threat that wouldn’t be believed, i.e. that threat was not a credible one. The concept of perfect equilibrium, developed by Selten (1965;1975), requires that the “strategies chosen by the players be a Nash equilibrium, not only in the game as a whole, but also in every subgame of the game”. (In our model on the picture, the subgame starts with the word “incumbent”). We have got the perfect equilibrium to rule out the undesirable one.

4. Repeated games approach.

a.) 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:

1. Whether the game is repeated infinitely or there is some finite number of times;

2. Whether there is a full information available to each firm about the objectives of, and opportunities available to, other firms;

3. How much weight the firms attach to the future in their calculations;

4. Whether the “cheating” can/can not be detected due to the knowledge/lack of knowledge about the prior moves of the firm’s 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 prisoner’s 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 it’s 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 rival’s 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, A’s 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 firm’s 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.