Game Theory and Business Strategy

Factors Affecting Sustainability of Cooperation

In this note, we will consider how the ability to collude under the grim trigger strategy is altered by these factors:

Market Concentration

The more firms in an industry, the more difficult it is to sustain cooperation.

Consider an industry with N symmetric firms. The firms can compete or collude. If the firms collude, they evenly divide the monopoly profit (the profit that would be earned if a single firm controlled the market). If a firm wishes to cheat, it can charge a price just slightly below the monopoly price and capture all of the consumers. The punishment after such a deviation is severe price competition, driving all firms' profits down to zero. Denote the monopoly profit by P.

if the firm: it earns:
cooperates P/N
cheats P
is being punished 0

As the number of firms rises (N increases), the profit from cheating doesn't change, and neither does the punishment. But, with more firms, each firm is earning less from cooperating, or colluding. So, the more firms in an industry, the more incentive there is to cheat, and the less likely that collusion can be sustained.

Difficulty of Detection

The harder it is to detect cheating, the harder it is to sustain cooperation.

In general, cooperation requires that the payoff stream from cheating is low enough. For example, in the above market, a firm's present value of profits from cheating is (r is the interest rate):

P + 0/(1+r) + 0/(1+r)2 + 0/(1+r)3 + ... = P

since after cheating, according to the grim trigger strategy, the firm will be punished forever.

However, say that it takes firms two periods to detect cheating. Then, I can "get away" with cheating for two periods, providing for a present value of:

P + P/(1+r) + 0/(1+r)2 + 0/(1+r)3 + ... = P + P/(1+r)

Obviously, the incentive to cheat is greater.

Extent of Consumer Lock In

The less consumer lock in, the more difficult it is to sustain cooperation.

If many consumers are "locked in" to a firm, then even if I cheat, I will not capture those consumers, so the incentive to cheat is small. However, by cheating and lowering its price, the firm is providing a discount to those consumers that are locked in to it, further lowering profitability.

At the extremes, consider an industry with no lock-in, so that every consumer always goes to the lowest price firm, and an industry in which almost every consumer is locked-in to some firm, effectively making each firm a monopolist. In the first industry, collusion is quite difficult since by cheating, a firm captures the entire market share, making the incentive to cheat large. In the second industry, cheating provides little benefit in terms of greater market share, but results in lower prices for the locked-in customers.


If the firms are not symmetric, so that some have a better product, more locked-in consumers, lower costs, etc., then the maximum interest rate to sustain cooperation will vary from firm to firm. In order to have cooperation, the actual interest rate has to be lower than every firm's maximum, in order to guarantee that no firm has incentive to cheat.


Tit-for-Tat may not be enough deterrence

Again consider the symmetric industry described above. Imagine that there are only two firms, the monopoly profit is 50, which is divided by both firms if they collude or captured by one firm if it cheats. Both firms employ Tit-for-Tat, so that after a period of cheating, both firms earn 0.

In order for a firm not to cheat, it must be:

25 + 25/(1+r) + 25/(1+r)2 + 25/(1+r)3 + ... > 50 + 0/(1+r) + 25/(1+r)2 + 25/(1+r)3 + ...

which reduces to: r<0. But this doesn't make sense, usually. For the interest rate to be less than 0, it means that firms have to "value the future more than they value the present." Or, you have to invest more than one dollar this year to have one dollar next year. The intuition for this is straightforward. The choice is to collude, and divide the profits every year, or to cheat and get all of the profits now and no profits next year. Given the option between getting all of the profits this year, or getting half this year and half next, who wouldn't take all of the profits now?

Thus, tit-for-tat is not enough of a deterrence in this simple duopoly collusion. As we saw earlier, adding more firms only makes things worse. So often, in business, more severe punishments are required. However, keep in mind the lessons from tit-for-tat. Any punishment should be forgiving, nice, provocable, and clear!