Strict Dominance in Game Theory 🎮

students, imagine you are choosing between two moves in a game, and one move is always better than another no matter what the other player does. That idea is called strict dominance. It is one of the most useful tools in game theory because it helps you quickly remove bad choices and focus on the strategies that could actually make sense.

What you will learn

By the end of this lesson, students, you will be able to:

define strict dominance,
detect strictly dominated strategies in examples,
explain why strictly dominated strategies can be removed from consideration.

This matters because real decisions often involve uncertainty. In a business choice, a sports play, or a classroom game, you want to avoid options that are always worse than another available option. Strict dominance gives a clear way to do that ✅

What strict dominance means

A strategy is strictly dominated if there is another strategy that gives a higher payoff every single time the opponent chooses an action.

More formally, suppose player $i$ has two strategies, $s_i$ and $t_i$. Strategy $s_i$ strictly dominates strategy $t_i$ if for every possible choice of the other players, the payoff from $s_i$ is greater than the payoff from $t_i$.

In symbols, if $u_i(\cdot)$ is player $i$'s payoff function, then $s_i$ strictly dominates $t_i$ when

$$u_i(s_i, s_{-i}) > u_i(t_i, s_{-i}) \text{ for every } s_{-i}.$$

Here, $s_{-i}$ means the strategies chosen by all players other than player $i$.

The key word is every. Not most of the time. Not usually. Every possible case. If one strategy is always worse, it should not be part of serious strategic thinking.

A simple real-world example 🥤

Imagine a cafeteria selling two drinks:

Drink A costs $2$ and gives you a medium cup,
Drink B costs $2$ and gives you a small cup.

If the price is the same, and Drink A gives you more drink in every situation, then Drink B is strictly dominated by Drink A. No matter what you value about the size of the drink, Drink B never does better.

In game theory, this same logic applies to strategic choices. If one action always gives a better payoff than another, the worse action is strictly dominated.

How to detect strict dominance in a payoff table

The easiest place to look for strict dominance is in a payoff matrix. A payoff matrix lists the outcomes for each combination of strategies.

Suppose player 1 has strategies $U$ and $D$, and player 2 has strategies $L$ and $R$. Player 1’s payoffs are shown below:

$\begin{array}{c|cc}$

& L & R \\

$\hline$

U & 4 & 3 \\

D & 2 & 1 \\

$\end{array}$

Compare player 1’s payoffs for $U$ and $D$:

if player 2 chooses $L$, then $U$ gives $4$ and $D$ gives $2$,
if player 2 chooses $R$, then $U$ gives $3$ and $D$ gives $1$.

Since $U$ gives a higher payoff in both cases, $U$ strictly dominates $D$. So $D$ is a strictly dominated strategy.

Notice that we did not need to guess what player 2 would do. Strict dominance checks all possible opponent actions at once.

Why strict dominance matters

If a strategy is strictly dominated, it is never the best choice for a rational player who wants the highest payoff.

Why? Because there is always another strategy that does better, no matter what happens. Choosing the dominated strategy means giving up payoff without any possible benefit. That makes the strategy easy to eliminate.

This is a form of reasoning called dominance reasoning. It is useful because it reduces the number of strategies you need to think about. In larger games, that can save a lot of time and make the analysis much clearer.

Another example with both players

Let’s look at a game where both players choose $A$ or $B$, and payoffs are listed as $(\text{player 1}, \text{player 2})$:

$\begin{array}{c|cc}$

& A & B \\

$\hline$

A & (2,2) & (1,3) \\

B & (3,1) & (0,0) \\

$\end{array}$

For player 1:

If player 2 plays $A$, then $B$ gives $3$ and $A$ gives $2$.
If player 2 plays $B$, then $A$ gives $1$ and $B$ gives $0$.

So neither $A$ nor $B$ strictly dominates the other for player 1, because each is better in one column.

For player 2:

If player 1 plays $A$, then $B$ gives $3$ and $A$ gives $2$.
If player 1 plays $B$, then $A$ gives $1$ and $B$ gives $0$.

Again, no strict dominance exists between the two strategies for player 2.

This example shows an important fact: not every game has strictly dominated strategies. Sometimes every strategy can survive the first check.

Strictly dominated strategies can still be removed 🚫

If a strategy is strictly dominated, we can remove it from the game because a rational player would never choose it.

This does not mean the game disappears. It means we simplify the game by deleting choices that are clearly inferior.

For example, suppose player 1 has three strategies $X$, $Y$, and $Z$, and $Z$ is strictly dominated by $X$. Then $Z$ can be ignored when we look for reasonable play. Any prediction that includes $Z$ as a likely choice is weaker than one that excludes it.

This is especially helpful when solving games step by step. After eliminating one dominated strategy, new dominated strategies may appear in the smaller game. That is one reason dominance reasoning is powerful.

Important detail: strict dominance is stronger than weak dominance

Strict dominance means the dominating strategy is better in every case. That is different from weak dominance, where a strategy is at least as good in every case and better in at least one.

For this lesson, focus on strict dominance. The rule is simple:

compare payoffs across all opponent strategies,
if one strategy gives a strictly higher payoff every time, the other strategy is strictly dominated.

Because the inequality is strict, there are no ties allowed in the comparison.

A step-by-step method for finding strictly dominated strategies

students, here is a reliable method you can use:

List the strategies for one player.
Compare payoffs for two strategies at a time.
Check whether one strategy gives a higher payoff in every row or column.
If yes, the worse strategy is strictly dominated.
Remove it and repeat if needed.

Let’s use this method on a small table.

$\begin{array}{c|ccc}$

& L & M & R \\

$\hline$

X & 5 & 4 & 6 \\

Y & 3 & 2 & 4 \\

Z & 1 & 7 & 0 \\

$\end{array}$

Comparing $X$ and $Y$:

against $L$, $X=5$ and $Y=3$,
against $M$, $X=4$ and $Y=2$,
against $R$, $X=6$ and $Y=4$.

Since $X$ is higher in all three cases, $X$ strictly dominates $Y$. So $Y$ can be removed.

Now compare $X$ and $Z$:

against $L$, $X=5$ and $Z=1$,
against $M$, $X=4$ and $Z=7$,
against $R$, $X=6$ and $Z=0$.

Here, $X$ is not higher in every case because $Z$ does better against $M$. So $Z$ is not strictly dominated by $X$.

This shows why you must check every opponent action carefully.

Why rational players ignore strictly dominated strategies

A rational player chooses strategies to get the best possible payoff. If strategy $t_i$ is strictly dominated by $s_i$, then $t_i$ always yields less payoff than $s_i$.

So if a player believes the payoffs are correctly described, choosing $t_i$ would be a mistake. There is no scenario where $t_i$ beats $s_i$. That is why game theorists remove strictly dominated strategies when analyzing rational behavior.

This is also a foundation for more advanced ideas like rationalizability. If a strategy cannot be justified because it is always worse than another available option, it should not be part of the predicted strategic outcome.

Conclusion

Strict dominance is a simple but powerful idea in game theory. A strategy is strictly dominated when another available strategy gives a better payoff in every possible case. Because such a strategy is never the best response to anything, it can be removed from the game. This helps simplify strategic analysis and focus attention on choices that a rational player might actually use.

When you look at a payoff table, students, remember to ask: is one strategy always better than another? If the answer is yes, the worse strategy is strictly dominated and can be eliminated. That is a key first step in reasoning about strategic behavior ✅

Study Notes

Strict dominance means one strategy gives a higher payoff in every possible case than another strategy.
If strategy $s_i$ strictly dominates $t_i$, then $u_i(s_i, s_{-i}) > u_i(t_i, s_{-i})$ for every $s_{-i}$.
A strictly dominated strategy is never the best choice for a rational player.
In payoff tables, compare strategies across all opponent actions.
If one strategy always has a higher payoff, the other strategy is strictly dominated.
Strictly dominated strategies can be removed to simplify the game.
This process is a key part of dominance reasoning and helps narrow feasible strategic choices.