<p>This really depends on how one defines a &quot;large&quot; player. But for the Saudi Arabia/OPEC situation mentioned in the <a href="https://www.pbs.org/wgbh/americanexperience/features/nash-game/" rel="nofollow noreferrer">PBS article you linked</a> one can imagine a made-up numerical example like this:</p>
<p>Take an infinitely repeated game, and let the inverse demand for oil be  <span class="math-container">$P(Q) = 32 - Q$</span> in each time period, where <span class="math-container">$Q$</span> is the aggregate oil output in the world. Suppose there is a Large oil-producing country (<span class="math-container">$L$</span>) and ten small oil-producing countries (<span class="math-container">$1,2,\dots$</span>).
For simplicity's sake let the marginal cost of producing oil be <span class="math-container">$0$</span> for all countries, but they are constrained by their capacities: <span class="math-container">$L$</span> can produce up to <span class="math-container">$24$</span> units, while each small country can produce up to <span class="math-container">$4$</span> unit of oil.</p>
<p>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.</p>
<p><strong>Claim 1</strong> Everyone producing at capacity in each time period no matter what is a Nash-equilibrium.</p>
<p>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.</p>
<p><strong>Claim 2</strong> If the countries agree on a total production of <span class="math-container">$Q = 16$</span> (monopoly level) and achieve this by each of them producing at <span class="math-container">$25\%$</span> capacity, i.e.
<span class="math-container">$q_L = 6$</span> and <span class="math-container">$q_1 = q_2 = \dots = q_{10} = 1$</span>, 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.</p>
<p>As country <span class="math-container">$L$</span> 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.</p>

CC BY-SA 4.0
2021-10-06T11:58:22.543Z
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?
Nguyen Lis
CC BY-SA 4.0
2021-10-06T12:18:31.210Z
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](https://economics.stackexchange.com/questions/47840/an-effort-to-link-cartel-and-nash-equilibrium#comment77120_47840), you will get a more nuanced understanding of the topic.
Giskard
CC BY-SA 4.0
2021-10-06T12:21:04.327Z
[Here is the respective chapter of the Osborne-Rubinstein's textbook](https://arielrubinstein.tau.ac.il/books/GT.pdf#page=149), 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.
Giskard
CC BY-SA 4.0
2021-10-06T05:39:25.383Z
2021-10-06T12:16:35.593Z
2021-10-06T12:16:35.593Z
Giskard
<p>I am reading some <a href="https://www.pbs.org/wgbh/americanexperience/features/nash-game/" rel="nofollow noreferrer">summary introduction to get the idea of game theory</a>. I saw a sentence:</p>
<blockquote>
<p>A &quot;large&quot; player who suffers disproportionately more from complete
finking may act cooperatively even when the small fry are finking</p>
</blockquote>
<p>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 <a href="https://en.wikipedia.org/wiki/Tit_for_tat" rel="nofollow noreferrer">link</a></p>
<blockquote>
<p>The Tit for Tat strategy copies what the other player previously
choose. If players cooperate by playing strategy (C,C) they cooperate
forever.</p>
</blockquote>
<p>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.</p>

<game-theory>
Why do large players suffer more for complete finking in tit-for-tat strategy?
Nguyen Lis
# Why do large players suffer more for complete finking in tit-for-tat strategy?

(Asked by: username_0 on 2021-10-05)

I am reading some [summary introduction to get the idea of game theory](https://www.pbs.org/wgbh/americanexperience/features/nash-game/). 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](https://en.wikipedia.org/wiki/Tit_for_tat)

> 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.

**Tags:** game-theory

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## 1 answers

### Answer by username_1 on 2021-10-06. Score: 2 | Accepted answer

This really depends on how one defines a "large" player. But for the Saudi Arabia/OPEC situation mentioned in the [PBS article you linked](https://www.pbs.org/wgbh/americanexperience/features/nash-game/) 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.

> **Comments:**
> **username_0**: 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?  
> **username_1**: 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](https://economics.stackexchange.com/questions/47840/an-effort-to-link-cartel-and-nash-equilibrium#comment77120_47840), you will get a more nuanced understanding of the topic.  
> **username_1**: [Here is the respective chapter of the Osborne-Rubinstein's textbook](https://arielrubinstein.tau.ac.il/books/GT.pdf#page=149), 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.  

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