# Trouble Understanding Dominant Strategy in the Prisoners Dilemma

Here's a payoff matrix for a Prisoners Dilemma:

Alpha is supposed to be the dominant strategy, which means that "no matter what my pair chooses, I will be better off playing Alpha than playing Beta."

I'm having trouble understanding this, because if my pair plays Beta, I'm better off playing Beta than Alpha.

My question is: "Why is Alpha the dominant strategy?" (or "Why is defecting the dominant strategy in the prisoners dilemma?")

• I just noticed that this matrix doesn't follow the T>R>P>S criteria of the Prisoners Dilemma. Here it seems that R<P. But in this lecture, the teacher says that Alpha is the dominant strategy! So the question remains... youtube.com/watch?v=nM3rTU927io Aug 7, 2023 at 15:45

if my pair plays Beta, I'm better off playing Beta than Alpha.

In case of $$(\beta,\beta)$$, you get 1, but in case of $$(\alpha,\beta)$$, meaning you play $$\alpha$$ against pair's $$\beta$$, you get 3, and $$3>1.$$

• Oh lort, I misspoke. Let's see... if I play 𝛼 and my pair plays 𝛼, I get 0. But if I play 𝛽 and my pair plays 𝛽, I get 1. That makes me think that playing 𝛽 can be better than 𝛼, so 𝛼 can't be a strictly dominating strategy. Aug 7, 2023 at 19:45
• No matter what fixed strategy your partner plays, you are better off playing $\alpha$. This is all that the definition is. It does not say anything about what happens if both of you change strategies. Aug 7, 2023 at 19:55
• Ahh I get it, I wasn't looking at pair's strategy as having to be fixed. Thank you! Aug 8, 2023 at 6:06

Going by the definition of dominant strategy from here:

A strategy $$s_i^∗$$ strictly dominates $$s_i$$ iff $$u_i(s^∗_i , s_{−i}) > u_i(s_i, s_{−i})$$, $$∀s_{−i} \in S_{−i}$$.

Here we have $$S_{-me}=\{\alpha, \beta\}$$ and $$s_{me} \in \{\alpha, \beta\} = S_{me}$$, with

$$u_{me}(\alpha, \alpha) = 0 > -1 = u_{me}(\beta, \alpha)$$,

$$u_{me}(\alpha, \beta) = 3 > 1 = u_{me}(\beta, \beta)$$

Combining the above two,

$$u_{me}(\alpha , s_{−me}) > u_{me}(s_{me}, s_{−me})$$, $$∀s_{−me} \in S_{−me}$$ and $$s_{me} \in S_{me} - \{\alpha\}$$

$$\implies s^∗_{me} = \alpha$$, i.e., $$\alpha$$ is the dominant strategy for me.