Weakly Dominant Strategy¶

Definition¶

Let $S_i$ be player $i$'s strategy set and $S_{-i}$ the set of strategy profiles of the other players in a normal-form game with payoffs $v_i$. A strategy $s_i\in S_i$ weakly dominates $s_i'\in S_i$ if, for all $s_{-i}\in S_{-i}$$S_{-i}$,

with strict inequality for at least one $s_{-i}\in S_{-i}$$S_{-i}$. The strategy $s_i$ is weakly dominant if it weakly dominates every other strategy $s_i'\ne s_i$ of player $i$. In a Bayesian game the same definition is applied type by type: for each type, the action must do at least as well against every action profile of the opponents, and strictly better against at least one.

Interpretation¶

A weakly dominant strategy is never worse than any alternative, whatever the others do, and is strictly better against at least one of their choices. A player who uses it needs no beliefs about the opponents: it is a best response to every $s_{-i}$.

In the figure (the truthful-bidding example below), a bid that beats the highest rival bid earns $10$ minus that rival bid, and a losing bid earns $0$. Bidding $10$ and bidding $8$ give the same payoff when the highest rival bid is below $8$ (for example $3$ at $7$) or above $10$ (for example $0$ at $12$), and bidding $10$ is strictly better in between ($1$ versus $0$ at $9$). That is weak dominance: never worse, and strictly better against some rival bids.

The difference from a strictly dominant strategy is that ties are allowed. Strict dominance requires a strictly higher payoff against every $s_{-i}$; weak dominance only requires "never worse, sometimes better". Every strictly dominant strategy is therefore weakly dominant, but not conversely.

Ties have consequences. A weakly dominated strategy can still be a best response to some opponent profile, so it can appear in a Nash equilibrium, and iterated deletion of weakly dominated strategies can give different results depending on the order of deletion. A leading application is the second-price auction, where bidding one's own value is weakly (not strictly) dominant.

Key Results¶

• Strictly dominant $\Rightarrow$ weakly dominant; the converse fails (see the example below).
• Under this definition a player has at most one weakly dominant strategy: two strategies cannot each weakly dominate the other, because each would need a strict inequality the other rules out.
• If every player has a weakly dominant strategy, the profile of these strategies is a Nash equilibrium. It need not be the only Nash equilibrium.
• A Nash equilibrium may involve weakly dominated strategies; it can never involve strictly dominated ones.
• In a second-price sealed-bid auction with private values, bidding $b_i=\theta_i$ (one's valuation) is weakly dominant; in a first-price auction, with at least two bidders, a positive-value type has no weakly dominant bid against unrestricted nonnegative rival bids. A zero-value type is an exception: bid $0$ always earns $0$ and weakly dominates every positive bid.

Examples¶

A weakly but not strictly dominant strategy. Consider

For the row player, $U$ gives $1>0$ against $L$ and $0=0$ against $R$, so $U$ weakly dominates $D$ and is weakly dominant, but not strictly dominant. The same holds for $L$ for the column player. Both $(U,L)$ and $(D,R)$ are Nash equilibria: at $(D,R)$ neither player gains by deviating alone, although both use weakly dominated strategies. Deleting weakly dominated strategies selects $(U,L)$.

Truthful bidding. A bidder with value $10$ in a second-price auction compares bidding $10$ with bidding $8$. If the highest rival bid is $9$, bidding $10$ wins and gives $10-9=1$$1$, while bidding $8$ loses and gives $0$. If the highest rival bid is $7$ or $12$, both bids give the same payoff ($3$ and $0$). So bidding $10$ is never worse and sometimes strictly better.

Prerequisites¶

• Strictly Dominant Strategy
• Normal-Form Game
• Best Response

Related Concepts¶

• Second-Price Auction
• English Auction
• Strictly Dominated Strategy
• Nash Equilibrium

Sources¶

• Game Theory · Week 6

Aliases: weakly dominant, weakly dominates Keywords: weak dominance, weakly dominated strategy, dominant-strategy equilibrium, truthful bidding Symbols: element of, theta

Last updated: 27 October 2026 11:52