# Prisoner's dilemma as a Bayesian one-shot game

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.

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.

• If the types don't affect the payoffs, then there's no change in the predicted outcome. – Herr K. Feb 5 '19 at 16:11
• I found this blog related to one-shot Prisoner's dilemma online. schneier.com/blog/archives/2013/05/one-shot_vs_ite.html – Mike J Feb 5 '19 at 19:13
• @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 Feb 5 '19 at 20:54
• @johnny09: Yes. But how do types affect payoffs in your case? How do $P,R,S,T$ change when $\theta$ changes? – Herr K. Feb 6 '19 at 1:20
• @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. Feb 7 '19 at 4:52