Clarifying the example of large firms tilt to cooperate after shocks of competition in game theory?

I am reading the example of large firms' operation when finking option is given here.

A "large" player who suffers disproportionately more from complete finking may act cooperatively even when the small fry are finking. Thus Saudi Arabia acts as a swing producer in OPEC, cutting its output to keep prices high when others produce more; and the United States bears a disproportionate share of the costs of its military alliances. Finally, if the group as a whole will do better in its external relations if it enjoys internal cooperation, then the process of biological or social selection may generate instincts or social norms that support cooperation and punish cheating.

Regarding the last sentence, I understand that if the firms already colluded together, in a more competition market, they will enjoy continue their cooperation because the cooperation has been established and maintained for a long time( process of social selection may generate instincts or social norms that support cooperation and punish cheating)

However, I cannot understand the example of Saudi Arabia and US as above, could you please explain it to me?

This may be easy to understand in a "public good game" context. (The examples are rather about club goods, but the logic is the same.)

A "public good game" can be thought of as a generalized $$n$$-player prisoner's dilemma.

In a public good game there are $$n$$ players, each of whom have $$K$$ dollars.

The players make a simultaneous decision how much of their wealth they wish to invest in the public good. Let us denote the investment of player $$i$$ by $$x_i$$. Finally, each player collects a payoff of $$K - x_i + r \cdot \sum_i x_i,$$ where $$r$$ is a parameter between $$1/n$$ and $$1$$, meaning

1. ($$r < 1$$) any player is best off individually if they decrease their investment to 0.
2. ($$1/n < r$$) if all players increase their investment by the same amount, they are all better off.

In a one-shot game rational players will invest 0, even though everyone investing $$K$$ would be a Pareto-improvement. (Prisoner's dilemma type situation.)

In a repeated game context cooperation (non-zero investment) can be beneficial. If $$r$$ is large enough, it is possible that it is worthwhile to maintain cooperation even if some of the players defect, i.e. invest zero. E.g., if $$n = 5$$, $$r = \frac{2}{5}$$, and it is common knowledge that players 1 and 2 are meanies who will always invest 0, players 3,4,5 are still better off if the three of them always invest $$K$$ and not 0, as $$K - K + \frac{2}{5} \cdot (0 + 0 + K + K + K) = \frac{6}{5} \cdot K$$ while $$K - 0 + \frac{2}{5} \cdot (0 + 0 + 0 + 0 + 0) = K.$$

Now imagine a history where players 1 and 2 have always invested 0 dollars while players 3,4 and 5 have always invested $$K$$ dollars. If player 3 were to defect, that is they will from now on also invest 0 dollars in each time periods, this might prompt players 4 and 5 to decrease their investment as well. So in a sense player 3 is a "pivotal" player in this situation. (Note: so are players 4 and 5, since investment is needed from at least three players for the benefits to outweigh the costs.)

In the story described above the players were symmetrical; everyone's investment had the same effect and everyone had the same capacity to invest. The payoff function was also linear. If you do away with some of these assumptions, it is possible that a "large" player would become "pivotal" to the cooperation, that is without them it is very difficult for the others to maintain beneficial cooperation.

Note that these models/explanations simplify the real world situation to an annoying extent, e.g., they exclude communication, side deals and non-financial incentives.

Personally I also don't like it when biological selection theories are pushed onto institutions such as nations, as modern nations are very young by evolutionary standards and the replicator dynamics differ vastly from those in biology. IMO these are somewhat lazy "solutions"/theories/pearls of wisdom in search of a problem (which has more funding than the similarly easy ones in biology).

• a very nice and comprehensive explanation, tks @Giskard Oct 6 '21 at 11:25