Why Compare Payoffs Across Every Opponent Action?
In a strategic-form game, a player’s payoff from an action can depend on what the other player does. A choice that gives a high payoff against one opponent action may give a low payoff against another. Dominance provides a precise test for whether one action is at least as good as another for every possible opponent action. This is useful because it separates a robust payoff comparison from a guess about what the opponent will choose.
A payoff matrix records the outcomes for each combination of actions. Unless stated otherwise, payoff pairs below are written as (row player’s payoff, column player’s payoff). To test a row player’s strategies, compare the first number in each relevant cell. To test a column player’s strategies, compare the second number. The units may represent profit, utility, or another measure of preference; in each example, higher payoff is preferred.
These terms describe different roles. “Dominant” describes a strategy that compares favorably with all of that player’s alternatives. “Dominated” describes a strategy that compares unfavorably with at least one other strategy. A player can have a dominated strategy without having any dominant strategy. A strategy can also be dominated by more than one alternative.
For strict dominance, every comparison must be a strict gain. For weak dominance, no comparison may be a loss, and at least one must be a strict gain. If two strategies give exactly the same payoff against every opponent action, neither weakly dominates the other under this definition: there is no strict gain in any case.
A Systematic Payoff-Comparison Test
For a player with strategies \(s\) and \(t\), let \(u(s,a)\) be that player’s payoff when they choose \(s\) and the opponent chooses action \(a\). The comparison is made separately for every opponent action. The payoff differences \(u(s,a)-u(t,a)\) show whether \(s\) is better, equal, or worse than \(t\) in each case.
This test also explains why it is not enough to compare average payoffs across opponent actions. Dominance is a comparison conditional on each possible action, not a comparison of averages that depends on how likely each action is. Unless a probability distribution over the opponent’s actions is supplied and the question asks for expected payoffs, averaging does not answer the dominance question.
Identify which player’s strategy is being assessed. In a payoff pair, select that player’s payoff coordinate.
Compare one candidate strategy with another strategy available to the same player.
Record whether the candidate gives a higher, equal, or lower payoff in each case. Do not omit an action because it seems unlikely.
All strict gains imply strict dominance. No losses and at least one strict gain imply weak dominance. A loss in any case rules out both forms of dominance for that comparison.
Worked Examples
Worked Example: Testing for a Strictly Dominant Strategy
A hypothetical supplier is deciding whether to reserve capacity early or wait. The buyer can place either a standard order or a rush order. The payoff matrix, in thousands of dollars, is:
| Supplier’s action | Standard order | Rush order |
|---|---|---|
| Reserve early | (12, 6) | (8, 9) |
| Wait | (7, 5) | (6, 8) |
Method. Assess the supplier’s action by comparing the first payoff in each cell. Against a standard order, reserving early yields 12 and waiting yields 7, a difference of \(12-7=5\). Against a rush order, reserving early yields 8 and waiting yields 6, a difference of \(8-6=2\).
Both differences are strictly positive, so Reserve early strictly dominates Wait. Since these are the supplier’s only two actions, Reserve early is strictly dominant for the supplier. This conclusion does not depend on which order the buyer places.
Common mistake avoided. The buyer’s payoff is not part of this comparison. For example, the buyer receives 9 rather than 8 in the rush-order column when the supplier reserves early rather than waits, but that does not establish anything about the supplier’s preference. The supplier’s first payoff coordinate is the relevant one.
Worked Example: Identifying Weak Dominance When One Payoff Ties
A hypothetical insurer can offer a standard renewal process or a streamlined process. A broker chooses between submitting a complete file and submitting a file that needs follow-up. The insurer’s payoff is the first number in each cell.
| Insurer’s action | Complete file | Follow-up needed | Late file |
|---|---|---|---|
| Standard process | (6, 4) | (5, 7) | (2, 6) |
| Streamlined process | (6, 5) | (8, 6) | (4, 5) |
Method. Compare the insurer’s payoffs from Streamlined process with those from Standard process, column by column:
- Complete file: \(6-6=0\), so the payoffs tie.
- Follow-up needed: \(8-5=3\), so the streamlined process is better by 3.
- Late file: \(4-2=2\), so the streamlined process is better by 2.
There is no loss in any case, and there are strict gains in two cases. Therefore, Streamlined process weakly dominates Standard process. Because it does not give a strictly higher payoff against the complete file, it does not strictly dominate Standard process. With only these two insurer actions, Streamlined process is weakly dominant, not strictly dominant.
Common mistake avoided. Calling the streamlined process strictly dominant because it wins in two of three columns ignores the tie in the first column. Strict dominance requires a strict gain in every column, not merely in most columns.
Worked Example: Finding Dominated Actions Without Assuming a Dominant One
A firm considers three contract offers, \(A\), \(B\), and \(C\). A rival can choose one of three responses, \(X\), \(Y\), or \(Z\). Payoffs are hypothetical and in thousands of dollars.
| Firm’s offer | Rival chooses X | Rival chooses Y | Rival chooses Z |
|---|---|---|---|
| A | (6, 2) | (3, 5) | (4, 1) |
| B | (5, 3) | (4, 4) | (2, 2) |
| C | (4, 1) | (2, 3) | (1, 0) |
Step 1: Compare the firm’s offers. The firm’s payoffs for \(A\), \(B\), and \(C\) are the first coordinates. Comparing \(A\) with \(C\), the differences are \(6-4=2\), \(3-2=1\), and \(4-1=3\). Thus, \(A\) strictly dominates \(C\). Comparing \(B\) with \(C\), the differences are \(5-4=1\), \(4-2=2\), and \(2-1=1\). Thus, \(B\) also strictly dominates \(C\). The dominated offer \(C\) has two strict dominators.
Step 2: Check whether the firm has a dominant offer. Comparing \(A\) with \(B\), the differences are \(6-5=1\), \(3-4=-1\), and \(4-2=2\). Offer \(A\) loses to \(B\) against response \(Y\), so \(A\) does not dominate \(B\). Conversely, \(B-A\) gives \(-1\), \(1\), and \(-2\), so \(B\) does not dominate \(A\). Offer \(C\) cannot dominate either, since it has a lower payoff than both \(A\) and \(B\) in each column. Therefore, the firm has no dominant offer, even though \(C\) is dominated.
Step 3: Check the rival’s actions separately. The rival’s payoffs are the second coordinates. For each firm offer, compare response \(Y\) with response \(X\): against \(A\), \(5>2\); against \(B\), \(4>3\); and against \(C\), \(3>1\). So \(Y\) strictly dominates \(X\) for the rival. Comparing \(Y\) with \(Z\), the rival’s payoffs are \(5>1\), \(4>2\), and \(3>0\), respectively. Thus, \(Y\) also strictly dominates \(Z\). With only these three rival actions, \(Y\) is strictly dominant for the rival.
Common mistake avoided. A payoff pair is not a single payoff to both players. The firm’s comparisons use first coordinates, while the rival’s comparisons use second coordinates. The two players can have different dominance conclusions in the same matrix.
Why the Test Works—and What It Does Not Say
The comparison follows directly from what “better for every opponent action” means. Suppose the player is considering \(s\) instead of \(t\). If the opponent chooses some action \(a\), the player receives \(u(s,a)\) from \(s\) and \(u(t,a)\) from \(t\). The difference \(u(s,a)-u(t,a)\) is positive exactly when \(s\) gives a higher payoff in that case. Requiring this difference to be positive for every \(a\) proves strict dominance. Requiring it to be nonnegative in every case, with at least one positive difference, proves weak dominance.
The word “every” is essential. If a candidate beats an alternative against two opponent actions but loses against a third, neither strict nor weak dominance is established. A comparison also concerns one player at a time: the other player’s payoff cannot substitute for the payoff of the player whose decision is being assessed.
Dominance is a statement about a player’s payoffs across possible actions. A dominant strategy, if one exists, is preferable to that player’s other listed actions no matter what the opponent does. But the existence of a dominant strategy for one player does not by itself identify what the other player will choose or determine a complete outcome. Similarly, finding a dominated strategy tells us how that action compares with another; it does not mean that every other action is dominant.
Common Exam Traps
- Checking only one column. A strategy that pays more against one opponent action has not been shown to dominate. Complete every column comparison.
- Treating a tie as a strict gain. Equality is allowed under weak dominance, but it prevents strict dominance for that opponent action.
- Accepting a loss under weak dominance. Weak dominance permits ties, not losses. One lower payoff is enough to rule it out.
- Comparing the wrong player’s payoff. Use the row player’s payoff to assess row strategies and the column player’s payoff to assess column strategies.
- Concluding that a dominant strategy must exist. Some games have no strategy that dominates every alternative. Strategies can be incomparable because each does better in different cases.
- Confusing “dominated” with “not dominant.” Not being dominant does not establish that an action is dominated. To call it dominated, show a particular alternative that performs at least as well in every case and strictly better in at least one case (or strictly better in every case for strict domination).
Check Your Understanding
For each question, assess only the player named and compare payoffs across every listed opponent action.
- A row player chooses \(A\) or \(B\), while the column player chooses \(L\) or \(R\). The row payoffs are \(A:(7,4)\) and \(B:(5,4)\). Does \(A\) strictly or weakly dominate \(B\)? Is \(A\) dominant if these are the row player’s only actions? (Answer: \(A\) weakly dominates \(B\), not strictly; \(A\) is weakly dominant.)
- A row player has actions \(A,B,C\), with payoffs across opponent actions \(L,R\): \(A:(3,8)\), \(B:(4,6)\), \(C:(2,5)\). Identify any strictly dominated strategy and state whether the row player has a dominant strategy. (Answer: \(C\) is strictly dominated by both \(A\) and \(B\); there is no dominant strategy, since \(A\) and \(B\) each outperform the other in one column.)
- A column player chooses \(X,Y,Z\). The column player’s payoffs, listed against row actions \(U,D\), are \(X:(2,3)\), \(Y:(4,5)\), and \(Z:(1,0)\). Identify the column player’s dominant strategy and classify the other two actions. (Answer: \(Y\) is strictly dominant; both \(X\) and \(Z\) are strictly dominated by \(Y\).)
- A row player has actions \(M,N\). Against opponent actions \(P,Q,R\), their payoffs are \(M:(5,7,3)\) and \(N:(5,6,4)\). Does \(M\) dominate \(N\), does \(N\) dominate \(M\), or are they incomparable? (Answer: They are incomparable: \(M\) ties against \(P\) and wins against \(Q\), but loses against \(R\); \(N\) therefore also fails to dominate \(M\) because it loses against \(Q\).)
- A row player chooses \(R\) or \(S\), and the opponent chooses \(J,K,L\). The row payoffs are \(R:(9,6,2)\) and \(S:(8,6,3)\). Classify each strategy as strictly or weakly dominant/dominated, if applicable. (Answer: Neither strategy dominates the other: \(R\) wins against \(J\), ties against \(K\), and loses against \(L\); \(S\) loses against \(J\), ties against \(K\), and wins against \(L\). Neither is dominant or dominated.)