<p>What happens if we assume that there is incomplete information to the prisoner's dilemma game?</p>

<p>For example, suppose we have the following matrix with the utilities <span class="math-container">$T&gt;R&gt;P&gt;S$</span> and <span class="math-container">$2R&gt;S+T$</span> with only <span class="math-container">$2$</span> players and <span class="math-container">$2$</span> actions.</p>

<p><a href="https://i.sstatic.net/k9VZf.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/k9VZf.png" alt="enter image description here"></a></p>

<p>With <span class="math-container">$\Theta_i=\{\underline{\theta},\bar{\theta}\}$</span>, does this mean that we have <span class="math-container">$2$</span> subgames (ex-interim) where we fix player <span class="math-container">$1$</span>'s type and assign probabilities to player <span class="math-container">$2$</span>'s types? Is this is the correct way to analyze the game?</p>

<p>If there is a paper studying the one-shot Prisoner's dilemma as a Bayesian game, I'd appreciate it very much if you share it with me!</p>

<p>Thank you.</p>

CC BY-SA 4.0
2019-02-05T16:11:12.800Z
If the types don't affect the payoffs, then there's no change in the predicted outcome.
Herr K.
CC BY-SA 4.0
2019-02-05T19:13:41.240Z
I found this blog related to one-shot Prisoner's dilemma online.
https://www.schneier.com/blog/archives/2013/05/one-shot_vs_ite.html
Mike J
CC BY-SA 4.0
2019-02-05T20:54:25.457Z
@HerrK. But usually in a Bayesian game, we define the utility functions as $u:\Theta\times A\to\mathbb{R}$, where $\Theta$ is the type space and $A$ is the action space. My point is that the types will necessarily affect the payoffs, right?
johnny09
CC BY-SA 4.0
2019-02-06T01:20:42.473Z
@johnny09: Yes. But how do types affect payoffs in your case? How do $P,R,S,T$ change when $\theta$ changes?
Herr K.
CC BY-SA 4.0
2019-02-06T01:48:51.370Z
@HerrK. Maybe I didn't write my question precisely, but I am more worried about the equilibrium analysis of such a game where types affect the utilities, eg we have $(R+\underline{\theta},R+\bar{\theta})$ instead of simply $(R,R)$.
johnny09
CC BY-SA 4.0
2019-02-06T03:13:39.870Z
@johnny09: I'm not sure what you're worried about. Equilibrium in Bayesian games like the one you propose is a well understood concept. Players mutually best respond to each other's strategy in the presence of uncertainty. Perhaps you should be more precise about what your problem is.
Herr K.
CC BY-SA 4.0
2019-02-06T21:39:43.120Z
@HerrK.Would it be possible if you share a reference justifying your claim "well-understood concept"? As I wrote in my question, I don't know how to analyze the game so, I am interested in seeing a rigorous solution.
johnny09
CC BY-SA 4.0
2019-02-07T04:52:08.747Z
@johnny09: Sorry I didn't get pinged for your last comment. The graduate micro text by Mas-Colell, Whinston and Green is a good reference. Bayesian games and the associated Bayesian Nash equilibrium is introduced on page 255.
Herr K.
<game-theory><nash-equilibrium><bayesian-game>
Prisoner's dilemma as a Bayesian one-shot game
johnny09
# Prisoner's dilemma as a Bayesian one-shot game

(Asked by: username_0 on 2019-02-04)

What happens if we assume that there is incomplete information to the prisoner's dilemma game?

For example, suppose we have the following matrix with the utilities $T>R>P>S$ and $2R>S+T$ with only $2$ players and $2$ actions.

[![enter image description here](https://i.sstatic.net/k9VZf.png)](https://i.sstatic.net/k9VZf.png)

With $\Theta\_i=\{\underline{\theta},\bar{\theta}\}$, does this mean that we have $2$ subgames (ex-interim) where we fix player $1$'s type and assign probabilities to player $2$'s types? Is this is the correct way to analyze the game?

If there is a paper studying the one-shot Prisoner's dilemma as a Bayesian game, I'd appreciate it very much if you share it with me!

Thank you.

**Tags:** game-theory, nash-equilibrium, bayesian-game
> **Comments:**
> **username_1**: If the types don't affect the payoffs, then there's no change in the predicted outcome.  
> **username_2**: I found this blog related to one-shot Prisoner's dilemma online. https://www.schneier.com/blog/archives/2013/05/one-shot_vs_ite.html  
> **username_0**: @HerrK. But usually in a Bayesian game, we define the utility functions as $u:\Theta\times A\to\mathbb{R}$, where $\Theta$ is the type space and $A$ is the action space. My point is that the types will necessarily affect the payoffs, right?  
> **username_1**: @johnny09: Yes. But how do types affect payoffs in your case? How do $P,R,S,T$ change when $\theta$ changes?  
> **username_0**: @HerrK. Maybe I didn't write my question precisely, but I am more worried about the equilibrium analysis of such a game where types affect the utilities, eg we have $(R+\underline{\theta},R+\bar{\theta})$ instead of simply $(R,R)$.  
> **username_1**: @johnny09: I'm not sure what you're worried about. Equilibrium in Bayesian games like the one you propose is a well understood concept. Players mutually best respond to each other's strategy in the presence of uncertainty. Perhaps you should be more precise about what your problem is.  
> **username_0**: @HerrK.Would it be possible if you share a reference justifying your claim "well-understood concept"? As I wrote in my question, I don't know how to analyze the game so, I am interested in seeing a rigorous solution.  
> **username_1**: @johnny09: Sorry I didn't get pinged for your last comment. The graduate micro text by Mas-Colell, Whinston and Green is a good reference. Bayesian games and the associated Bayesian Nash equilibrium is introduced on page 255.  

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