# Why do large players suffer more for complete finking in tit-for-tat strategy?

I am reading some summary introduction to get the idea of game theory. I saw a sentence:

A "large" player who suffers disproportionately more from complete finking may act cooperatively even when the small fry are finking

I understand that when the market becomes more competitive under the intervention of the government, the firms used to cooperate will continue cooperate due to this link

The Tit for Tat strategy copies what the other player previously choose. If players cooperate by playing strategy (C,C) they cooperate forever.

But I am wondering why large player suffers more from complete finking? I am wondering if there is any intuitive example for that, especially for relating to big firm as a large player.

This really depends on how one defines a "large" player. But for the Saudi Arabia/OPEC situation mentioned in the PBS article you linked one can imagine a made-up numerical example like this:

Take an infinitely repeated game, and let the inverse demand for oil be $$P(Q) = 32 - Q$$ in each time period, where $$Q$$ is the aggregate oil output in the world. Suppose there is a Large oil-producing country ($$L$$) and ten small oil-producing countries ($$1,2,\dots$$). For simplicity's sake let the marginal cost of producing oil be $$0$$ for all countries, but they are constrained by their capacities: $$L$$ can produce up to $$24$$ units, while each small country can produce up to $$4$$ unit of oil.

Suppose the countries choose their production quantities simultaneously in each time period, and the goal of each country is to maximize the time average/discounted sum of its oil revenues.

Claim 1 Everyone producing at capacity in each time period no matter what is a Nash-equilibrium.

Total production in this case is 64 units per period, and no matter how much a single country decreases production the total will be above 32 units, so the market price is always zero, thus there is no incentive to deviate.

Claim 2 If the countries agree on a total production of $$Q = 16$$ (monopoly level) and achieve this by each of them producing at $$25\%$$ capacity, i.e. $$q_L = 6$$ and $$q_1 = q_2 = \dots = q_{10} = 1$$, then with a low enough discount factor and a threat of reverting to maximum production in case of defection this is also a Nash-equilibrium.

As country $$L$$ has a larger market share than other countries, it suffers more (four times as much) if the cooperative situation described in Claim 2 collapses and the state of the market changes to the one described in Claim 1.

• Hi @Giskard. Thanks a heap for your detailed answer. However, I am curious about some points regarding the Claim 2. (1) qL should equal to 6 rather than 4 as you documented (24*25%=6). And (2) I do not understand why the Claim 2 is a Nash equilibrium? More clear, why the countries should not deviate from the 25% agreement from the initial stage? Oct 6 '21 at 11:58
• Hi @NguyenLis! You were right about $q_L$, I corrected it. The players do not want to deviate because everyone threatens that in case of deviation they will return production to full capacity, resulting in 0 revenue in all future time periods. Once again I recommend reading the repeated games chapter of a game theory textbook, you will get a more nuanced understanding of the topic. Oct 6 '21 at 12:18
• Here is the respective chapter of the Osborne-Rubinstein's textbook, it is made available online by the authors for free. You may find it too difficult; in this case I recommend looking for an intro to game theory textbook. Oct 6 '21 at 12:21