Weak Dominance in Game Theory
Introduction
students, in many games, players do not choose randomly. They look for strategies that can never be improved upon by another option. One powerful idea in game theory is dominance reasoning, which helps players rule out choices that are clearly worse. In this lesson, you will learn about weak dominance, a type of strategy comparison that says one action is never better and sometimes worse than another action. 🎯
By the end of this lesson, you should be able to:
- Define weak dominance.
- Identify weakly dominated strategies in a game.
- Compare weak dominance with strict dominance.
This topic matters because it helps simplify strategic situations like pricing, voting, negotiation, and competition. If a strategy is weakly dominated, a rational player may avoid it because another strategy does at least as well in every case and does better in some cases. That kind of reasoning is a key step toward understanding rationalizability.
What Weak Dominance Means
Suppose a player has two strategies, $s$ and $t$. Strategy $s$ weakly dominates strategy $t$ if two conditions are true:
- $s$ gives at least as high a payoff as $t$ against every possible action of the other player or players.
- $s$ gives a strictly higher payoff than $t$ against at least one possible action of the other player or players.
In symbols, if $u(s, a)$ is the payoff from choosing $s$ when the other player chooses action $a$, then $s$ weakly dominates $t$ when
$$u(s,a) \ge u(t,a) \text{ for all } a,$$
and there exists at least one action $a'$ such that
$$u(s,a') > u(t,a').$$
This means $t$ is never better than $s$, and sometimes it is worse. 🔍
A weakly dominated strategy is a strategy that is weakly dominated by some other strategy. Rational players often try to avoid such strategies because there is another choice that is never worse and sometimes better.
A Simple Example
Let’s use a small two-player game. Player 1 can choose $A$ or $B$. Player 2 can choose $L$ or $R$. Player 1’s payoffs are shown below:
| | $L$ | $R$ |
|------|-----|-----|
| $A$ | $3$ | $2$ |
| $B$ | $3$ | $1$ |
Here, compare $A$ and $B$ for Player 1:
- If Player 2 chooses $L$, both $A$ and $B$ give payoff $3$.
- If Player 2 chooses $R$, $A$ gives payoff $2$ while $B$ gives payoff $1$.
So $A$ gives at least as much as $B$ in every case, and strictly more in one case. Therefore, $A$ weakly dominates $B$.
Notice something important: $B$ is not strictly dominated by $A$, because strict dominance would require $A$ to be better in every case. But here, at $L$, the payoffs are equal. That is exactly why this is weak dominance rather than strict dominance.
This example shows why weak dominance can be useful. Even if a strategy is sometimes tied with another one, it may still be reasonable to eliminate the weaker choice if it is never better and sometimes worse. ✅
How to Identify Weakly Dominated Strategies
To check whether a strategy is weakly dominated, follow these steps:
1. Compare payoffs against every opponent action
Look at all possible choices the other player might make. A strategy can only weakly dominate another if it is at least as good in every one of those cases.
2. Check for equality in some cases
A weakly dominating strategy is allowed to tie the other strategy in some situations. That is a key difference from strict dominance.
3. Look for at least one strict improvement
There must be at least one opponent action where the weakly dominating strategy does better. If it only ties everywhere, then the two strategies are payoff-equivalent, not weak dominance.
4. Be careful with mixed strategies in advanced settings
In many games, a strategy can be weakly dominated by a mixed strategy, meaning a probability combination of other strategies. For this lesson, focus mainly on comparing pure strategies, but it is good to know that dominance reasoning can extend further.
A fast way to think about it is this: if one strategy is never better and sometimes worse, it is weakly dominated. 🧠
Weak Dominance vs. Strict Dominance
Weak dominance and strict dominance are closely related, but they are not the same.
Strict dominance
Strategy $s$ strictly dominates strategy $t$ if
$$u(s,a) > u(t,a) \text{ for all } a.$$
That means $s$ is better in every possible situation. There are no ties.
Weak dominance
Strategy $s$ weakly dominates strategy $t$ if
$$u(s,a) \ge u(t,a) \text{ for all } a,$$
and for some $a'$
$$u(s,a') > u(t,a').$$
That means $s$ is at least as good everywhere and better somewhere, but it may tie in some cases.
Main difference
The main difference is the possibility of ties. Strict dominance has no ties at all. Weak dominance allows ties, as long as there is at least one strict gain.
This difference matters because weak dominance is a bit less demanding than strict dominance. A strategy can fail to be strictly dominated and still be weakly dominated. In other words, weak dominance catches more dominated strategies than strict dominance does. ⚖️
Another Example: Why Ties Matter
Consider Player 1’s payoffs below:
| | $L$ | $R$ |
|------|-----|-----|
| $C$ | $4$ | $2$ |
| $D$ | $4$ | $1$ |
Compare $C$ and $D$:
- Against $L$, both give $4$.
- Against $R$, $C$ gives $2$ and $D$ gives $1$.
So $C$ weakly dominates $D$.
Now ask: does $C$ strictly dominate $D$? No, because at $L$ they are equal. This example shows that a strategy can be weakly dominated even when it is tied in one outcome. The tie does not protect it from weak dominance.
Real-world analogy: imagine two delivery plans. Plan $C$ arrives on time both when traffic is light and when traffic is heavy. Plan $D$ arrives on time when traffic is light, but late when traffic is heavy. Plan $C$ is never worse and sometimes better. That makes $D$ the weaker plan. 🚚
Why Weak Dominance Helps Narrow Choices
Game theory often tries to reduce a big list of strategies to a smaller, more sensible list. Weak dominance is one tool for doing this.
If a strategy is weakly dominated, a player might exclude it because there is another strategy that performs at least as well in every situation and better in some. This does not prove the player will never choose the weakly dominated strategy in real life, but it does show the strategy is not attractive under standard rational reasoning.
This is connected to rationalizability, which means a strategy can be justified as a best response to some beliefs about what others will do. If a strategy is weakly dominated, it is often a poor candidate for rational choice because there is another strategy that is always at least as good.
However, one important caution is that eliminating weakly dominated strategies can sometimes be more delicate than eliminating strictly dominated strategies. Strictly dominated strategies are easier to remove because they are worse in every case. Weakly dominated strategies may survive in some solution concepts, especially when order of elimination matters. So weak dominance is powerful, but it must be used carefully. 🧩
Quick Practice Example
Suppose Player 1 has strategies $X$, $Y$, and $Z$ with payoffs:
| | $M$ | $N$ |
|------|-----|-----|
| $X$ | $5$ | $3$ |
| $Y$ | $5$ | $2$ |
| $Z$ | $4$ | $4$ |
Which strategy is weakly dominated?
Compare $X$ and $Y$:
- Against $M$, both give $5$.
- Against $N$, $X$ gives $3$ and $Y$ gives $2$.
So $X$ weakly dominates $Y$.
Compare $Z$ with $X$:
- Against $M$, $Z$ gives $4$, which is less than $5$.
- Against $N$, $Z$ gives $4$, which is more than $3$.
Neither weakly dominates the other.
So $Y$ is weakly dominated by $X$, while $X$ and $Z$ are not weakly dominated by each other based on this table.
When solving problems, always compare every pair carefully. A strategy is weakly dominated only if there is another strategy that matches or beats it in every column and beats it in at least one. 🔎
Conclusion
Weak dominance is a way to compare strategies and eliminate choices that are never better and sometimes worse than another option. students, you should now know that a strategy $s$ weakly dominates another strategy $t$ when $s$ gives at least as high a payoff in every case and strictly higher payoff in at least one case. You should also be able to tell the difference between weak dominance and strict dominance: strict dominance requires a strict improvement everywhere, while weak dominance allows ties.
This idea is important because it helps simplify games and supports rational choice reasoning. By spotting weakly dominated strategies, you can narrow down the set of plausible actions and better understand strategic behavior. 💡
Study Notes
- Weak dominance means one strategy is never worse than another and sometimes better.
- If $s$ weakly dominates $t$, then $u(s,a) \ge u(t,a)$ for all $a$, and $u(s,a') > u(t,a')$ for at least one $a'$.
- A weakly dominated strategy is a strategy that another strategy weakly dominates.
- Strict dominance requires $u(s,a) > u(t,a)$ for all $a$.
- Weak dominance allows ties; strict dominance does not.
- To identify weak domination, compare payoffs against every possible action of the opponent.
- Weak dominance is useful for narrowing strategic choices and understanding rationalizability.
- Be careful: weak dominance is more delicate than strict dominance, especially when eliminating strategies step by step.