2.2b Strategic Dominance

Strategic Dominance

Strategic dominance in game theory occurs when there is a certain single strategy $s_i^{\prime }\in S_i$ for a player $i\in N$ that is the best response for any combination of other players’ strategies $s_{-i}$. There are two cases in which this happens:

• Strategy $s_i^{\prime }$ strongly dominates all other strategies $s_i$ of player $i$ if the payoff of $s_i^{\prime }$ is strictly greater than all of $i$‘s other strategies $s_i$ for some strategies of other players $s_{-i}$, such that
  $u_i(s_i^{\prime },s_{-i})>u_i(s_i,s_{-i}).$
• Strategy $s_i^{\prime }$ weakly dominates another strategy $s_i^{\#}\in S_i$ if it does at least as well as $s_i^{\#}$ for all possible strategies of other players $s_{-i}$, and does better than $s_i^{\#}$ for certain strategies ${\hat{s}}_{-i}$ of the other players, such that
  \begin{equation}\begin{aligned}u_i(s_i', s_{-i}) &\geq u_i(s_i^\#, s_{-i}) && \text{for all $s_{-i}$} \\ u_i(s_i', \hat{s}_{-i}) &> u_i(s_i', \hat{s}_{-i}) && \text{for some } \hat{s}_{-i}\end{aligned}\end{equation}

A strategy profile to a game in which each player is using their dominant strategy is said to be a dominant strategy solution.

Intuition

Strategic dominance follows the principle of rational choice. Obviously, considering the other players’ strategies, one will choose that strategy which puts them at the best position possible.

• Strongly dominant strategies can be interpreted as strategies that are the only rational response of a player $i$ in a game.
• Weakly dominant strategies can be interpreted as a set of rational strategies that can be used as a best response depending on the other players’ strategies.
  Here, it is important to note that not all games have dominant strategies.

Let’s now look at the solvability of dominance. pg 69-71 (Optional?)

Examples

EASY: Dominant Strategy in Auctions

Suppose you want to buy a painting of Renoir. You value it at $300,000,meaningthisisthetruevaluationthatyougivethepainting.Then,therulesofgametheorydictatethatactivelybiddinguntil$300,000 and either giving up or winning the auction is the dominant strategy. Consider the other two cases:

• Suppose you set yourself a lowball limit of $250,000.Then,biddingeithercontinuesaboveyourtruepriceof$300,000, or somebody wins the auction with a bid of $270,000. In that case, you’re bummed, since you could’ve gotten a sweet painting for less than it’s worth to you.
• Suppose now you overextended yourself and decide to bid until $350,000. Then, either somebody saves you by bidding higher, or you win the auction. However, you’re still bummed, since you are now stuck with a painting that you paid more for than you think is worth.

MEDIUM: Electing the UN Secretary-General

See pg. 68