# Which pure strategies in each game are dominated?

Which (pure strategies) in each game are dominated? For each dominated strategy specify the (mixed) strategy that dominates it.

The solution manual says that

in Game 1, R is dominated by $$\sigma_2 = (1/2, 1/2, 0)$$

and B is dominated by $$\sigma_1 = (1/2, 1/2, 0)$$

in Game 2, B is dominated by $$\sigma_1 = (3/5, 2/5, 0)$$

and M is weakly dominated by $$\sigma_2 = (1/4, 0, 3/4)$$

I don't understand this solution completely. And also, how these sigma values are written?

• In game 1, $\sigma_2=(1/2,1/2,0)$ mean a mixed strategy half L and half M, with payoffs for Player 2 of $3,2,2$ which dominates the payoffs for Player 2 from R of $1,1,1$ Apr 10, 2021 at 20:43

We will see these in order:

In Game 1, R is dominated by $$\sigma_2 = (1/2, 1/2, 0)$$

Notation: $$\sigma_2$$ is a mixed strategy for player-2 to play the choices $$L,M$$ with probability $$\frac{1}{2}$$ each.

Explanation for dominance: $$\sigma_2$$ dominates R as in R, player-2 wins 1 irrespective of what player-1 plays (see the right side of the column 3 values on table 1). Thus the expected earnings for player-2 when player-1 plays strategies $$\{T,C,B\}$$ is $$\{1,1,1\}$$. On the other hand, if Player-2 plays $$\sigma_2$$, then the expected earning is $$\{\frac{4+2}{2},\frac{0+4}{2},\frac{3+1}{2}\} = \{3,2,2\}$$ when player-1 plays $$\{T,C,B\}$$.

Note that $$\{3,2,2\}$$ dominates $$\{1,1,1\}$$. Therefore, irrespective of what player-1 plays, player-2 should play $$\sigma_2$$. Thus, $$\sigma_2$$ dominates R

In Game 1, B is dominated by $$\sigma_1 = (1/2, 1/2, 0)$$ The expected payoff to player-1 under B when player-2 plays $$\{L,M,R\}$$ is $$\{1,1,1\}$$ The expected payoff to player-1 under $$\sigma_1$$ when player-2 plays $$\{L,M,R\}$$ is $$\{\frac{4+2}{2},\frac{0+4}{2},\frac{3+1}{2}\} = \{3,2,2\}$$, which dominates $$\{1,1,1\}$$. Thus $$\sigma_1$$ dominates B.

In Game 2, B is dominated by $$\sigma_1 = (3/5, 2/5, 0)$$

The expected payoff to player-1 under B when player-2 plays $$\{L,M,R\}$$ is $$\{2,2,2\}$$ The expected payoff to player-1 under $$\sigma_1= (3/5, 2/5, 0)$$ when player-2 plays $$\{L,M,R\}$$ is $$\{\frac{3}{5}\times 0 + \frac{2}{5}\times 6,\frac{3}{5}\times 3 + \frac{2}{5}\times 1,\frac{3}{5}\times 5 + \frac{2}{5}\times 6\} = \{\frac{12}{5},\frac{11}{5},\frac{27}{5}\}$$, which dominates $$\{2,2,2\}$$. Thus $$\sigma_1$$ dominates B.

In Game 2, M is weakly dominated by $$\sigma_2 = (1/4, 0, 3/4)$$

The expected payoff to player-2 under M when player-1 plays $$\{T,C,B\}$$ is $$\{3,3,6\}$$ The expected payoff to player-2 under $$\sigma_2= (\frac{1}{4}, 0, \frac{3}{4})$$ when player-1 plays $$\{T,C,B\}$$ is $$\{\frac{1}{4}\times 2 + \frac{3}{4}\times 6,\frac{1}{4}\times 6 + \frac{3}{4}\times 2,\frac{1}{4}\times 0 + \frac{3}{4}\times 8\} = \{5,3,6\}$$, which weakly dominates $$\{3,3,6\}$$.

We say $$\sigma_2$$ weakly dominates M because (i) Payoff($$\sigma_2$$,Player-1 move) $$\geq$$ Payoff(M,Player-1 move) $$\forall$$ Player-1 moves, but (ii) $$\exists$$ Player-1 move such that Payoff($$\sigma_2$$,Player-1 move) $$>$$ Payoff(M,Player-1 move)

• Thank you for your great explanation. Then, for example, in the game 2 and the case of B, $\sigma_1=(3/5, 2/5,0)$ is chosen randomly? That’s, according to your explanation, when I calculate $\sigma_1=(p,1-p,0)$, I have found that $p\in (1/2, 2/3)$. That’s, can we choose any p value randomly in this range such that B is dominated?
– 1190
Apr 11, 2021 at 12:12
• yes, and then you may even have dominance between such strategies. ie given some $p_1, p_2$, find if $\sigma_1^1=(p_1,1-p_1,0)$ dominates $\sigma_1^2=(p_2,1-p_2,0)$. Optimizing over such simplices, one can find a single optimal (or an interval of equally optimal) strategies Apr 11, 2021 at 12:19
• Okay thank you. I clearly understand the answer thanks to you:)
– 1190
Apr 11, 2021 at 12:20
• Dear Madhavan, can you please look at my this question as well? I will be glad if you will make a clear explanation as it is. Thank you:) math.stackexchange.com/questions/4101269/…
– 1190
Apr 14, 2021 at 8:47