Week 1-Part 5: Dominant Strategies

Definition (Domination)

Let $s_i$ and $s'_i$ be two strategies (formally defined later, take it to mean :choice of action” ) for player $i$ and let $S_{-i}$ be the set of all possible strategy profiles for the other players. Then we say that:

• $s_i$ strictly dominates $s_{i}^{\prime}$ if $\forall s_{-i} \in S_{-i}, u_{i}\left(s_{i}, s_{-i}\right)>u_{i}\left(s_{i}^{\prime}, s_{-i}\right)$

  i.e. if, for every joint set of actions the other player will take, the utility player $i$ gets by playing $s_i$ is strictly more than the utility $i$ gets when they play $s^\prime _i$

• $s_i$ very weakly dominates $s_i^\prime$ if $\forall s_{-i} \in S_{-i}, u_{i}\left(s_{i}, s_{-i}\right) \geq u_{i}\left(s_{i}^{\prime}, s_{-i}\right)$

  i.e. if, for every joint set of actions the other player will take, the utility player $i$ gets by playing $s_i$ is greater than or equal to more than the utility $i$ gets when they play $s^\prime _i$

When a strategy dominates another strategy, you don’t need to think about what the other agents will do in order to decide that you would prefer to play $s_i$ instead of $s^\prime_i$. If one strategy dominates all others, we say it is dominant. If this is the case, then $s_i$ is better than everything else (i.e. it is the best thing to do). If you have a dominant strategy, you don’t need to worry about what the other agents will do, you can always play the dominant strategy.

Claim: A strategy profile consisting of dominant strategies for every player must be a Nash equilibrium. This must be true because since all the strategies are dominant, no agent will want to change their strategy. Furthermore, an equilibrium in a strictly dominant strategy must be unique. If not, it is not strictly dominant.

Example (Prisoner’s Dilemma)

Consider the following matrix:

$\begin{array}{c|c|c} & C & D \\ \hline C & -1,-1 & -4,0 \\ \hline D & 0,-4 & -3,-3 \\ \hline\end{array}$

Claim: Player 1 has a dominant strategy, which is playing $D$. We can prove this using a case analysis:

• Suppose Player 2 chooses $C$. Then, Player 1 faces a choice of getting a payoff of -1 or a payoff of 0. Since 0>1, player 1’s best response is to play $D$.
• Suppose Player 2 chooses $D$. Then, Player 1 faces a choice of getting a payoff of -4 or a payoff of -3. Since -3>-4, player 1’s best response is to play $D$.

Therefore, regardless to what Player 2 does, PLayer 1’s best response is to strictly choose $D$. Therefore $D$ is a dominant strategy. The same case analysis applies to Player 2 (symmetric)

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