How to Prove Two Finite Games Are Strategically Equivalent

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To prove that two finite normal-form games are strategically equivalent, first state which definition you are using. Under the payoff-difference definition, the proof is a finite check: after matching players and strategies, every player must have exactly the same payoff change from switching between any two of their strategies while the opponents’ profile is held fixed. A single unequal difference disproves equivalence. This relation preserves best responses and Nash equilibria, but not necessarily the numerical payoff levels.

What does strategically equivalent mean in game theory?

There is no universally safe use of the phrase strategically equivalent. In finite-game problems it commonly refers to at least four different comparisons:

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• Payoff-difference equivalence: every unilateral payoff difference is the same in the two games.
• Positive affine equivalence: one game’s payoffs are obtained from the other by positive affine transformations, sometimes with player or strategy relabeling.
• Strategic isomorphism: bijections match players and their strategies, and corresponding profiles satisfy a specified payoff-preservation rule.
• Response-graph equivalence: the games have isomorphic best-response or preference graphs.

These are not interchangeable tests. A matching equilibrium set or response graph does not by itself establish equality of all unilateral payoff differences. Before calculating anything, write down the convention required by your course, theorem, or assignment.

The payoff-difference definition

Let game G have players N, pure strategy sets S_i, and payoff functions u_i. Let game H have payoff functions v_i on the same labeled players and strategy sets. They are payoff-difference strategically equivalent when, for every player i, every pair of that player’s strategies s_i and t_i, and every fixed opponents’ profile s_−i,

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u_i(s_i, s_-i) - u_i(t_i, s_-i) = v_i(s_i, s_-i) - v_i(t_i, s_-i).

The opponents’ strategies are held constant. You compare what player i gains or loses by changing only their own action; you do not compare arbitrary cells or totals. Because every strategy set is finite, the definition produces a finite list of equalities that can be checked exhaustively.

What the definition allows

The two games need not display the same payoff numbers. For a fixed opponents’ profile, you may add a quantity that does not depend on the acting player’s own strategy. If

v_i(s_i, s_-i) = u_i(s_i, s_-i) + k_i(s_-i),

then the added term cancels when two own strategies are subtracted. The function k_i may vary with the opponents’ profile, so the absolute level of player i’s payoffs can change from one column, row, or opponents’ configuration to another.

Conversely, on identical strategy sets, the difference condition implies this form. Choose a reference strategy r_i for player i and define k_i(s_-i) = v_i(r_i,s_-i) - u_i(r_i,s_-i). Applying the defining equality to s_i and r_i gives v_i(s_i,s_-i)-u_i(s_i,s_-i)=k_i(s_-i) for every own strategy. Thus, the discrepancy between the payoff functions must be independent of the acting player’s action.

Set up the games before proving anything

List the players and pure strategies

Write each game as a complete strategic-form object:

• the player set;
• each player’s pure strategy set;
• the payoff assigned to every pure-strategy profile.

Do not infer equivalence from similar-looking matrices. A two-player matrix can have the same dimensions while representing different labels, action meanings, or player order. Record which coordinate belongs to which player and whether the entries are ordered as (player 1 payoff, player 2 payoff), or in another convention.

Decide whether relabeling is permitted

If the games use identical labels, compare corresponding players and strategies directly. If labels differ, state the proposed bijections before doing payoff arithmetic:

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• a player bijection maps each player in G to one player in H;
• for each matched player, a strategy bijection maps that player’s strategies in G to the corresponding strategies in H;
• the product of those strategy maps identifies every profile in one game with a profile in the other.

After relabeling, all comparisons must use the mapped profile. A failed comparison under one proposed matching rejects that matching, not necessarily every possible matching.

Step-by-step proof for payoff-difference equivalence

1. State the quantified criterion

Begin the proof with the exact condition rather than an informal claim that the games have the same incentives:

For all i, all s_i,t_i in S_i, and all s_-i in S_-i, u_i(s_i,s_-i)-u_i(t_i,s_-i) = v_i(s_i,s_-i)-v_i(t_i,s_-i).

If you are using a relabeling, replace each profile in the second game by its image under the stated bijections. If your assignment uses a different convention, write that transformation explicitly and do not switch to the difference test halfway through.

2. Enumerate the unilateral comparisons

For player i, choose every unordered pair of distinct own strategies. For each pair, hold each opponents’ profile fixed in turn. In a two-player matrix, this means comparing rows within every column for the row player and columns within every row for the column player. In a larger game, fix all opponents’ coordinates and vary only one player’s coordinate.

There are |S_-i| × C(|S_i|,2) comparisons for player i, where |S_-i| is the number of opponents’ profiles and C(|S_i|,2) is the number of unordered pairs of own strategies. You may check ordered pairs instead; the reverse equation then carries the opposite sign and adds no new information.

3. Subtract, do not merely compare rankings

For each fixed opponents’ profile, calculate the numerical difference in both games. Record the result in a table for a small game or show the algebra for a parameterized game. Saying that both games make action A preferable to action B is weaker: it proves only a ranking, not equality of the payoff change.

4. Conclude at the correct strength

If every required equality holds, conclude payoff-difference strategic equivalence under the stated player and strategy correspondence. If one equality fails, give that player, opponents’ profile, and pair of own strategies as a counterexample. Do not claim exact payoff equality unless you separately checked it; the definition allows profile-dependent additions independent of the acting player’s strategy.

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Worked 2×2 example

Consider two games with row player strategies A and B and column player strategies L and R. The entries are (row payoff, column payoff).

Game H was formed by adding 5 to the row player’s payoff in column L and −2 in column R. It was also formed by adding 7 to the column player’s payoff in row A and −4 in row B. Those additions depend on the opponent’s action, not on the acting player’s own action.

Row player comparisons

Column player comparisons

Every unilateral difference matches, so the games are strategically equivalent under this definition. Their displayed payoff levels differ substantially, but each player faces the same incentive to switch at every fixed opponent action. Consequently, the best-response correspondences and the pure and mixed Nash-equilibrium conditions are preserved under the identity mapping of profiles.

How to handle different labels or a player swap

Suppose game G has players 1 and 2, while game H labels them X and Y. A proposed correspondence might map 1 to Y, 2 to X, and map each strategy set by a stated bijection. Let φ denote the player map and ψ_i the strategy map for player i. A profile (s_1,...,s_n) in G is identified with (ψ_1(s_1),...,ψ_n(s_n)) in the appropriately reordered game H.

For payoff-difference equivalence, check, for every player i, own strategies s_i, t_i, and opponents’ profile s_−i, that the difference in G equals the difference for player φ(i) in H at the mapped profile:

u_i(s_i,s_-i) − u_i(t_i,s_-i) = v_φ(i)(ψ_i(s_i),ψ_-i(s_-i)) − v_φ(i)(ψ_i(t_i),ψ_-i(s_-i)).

For a stronger strategic-isomorphism claim, the required condition may instead demand exact equality of corresponding payoffs, or equality after a specified transformation. State that requirement separately. A bijection of labels alone proves nothing about incentives.

How the main notions differ

A positive affine transformation is not automatically the payoff-difference definition. If v_i = a_i u_i + b_i with a_i > 0, then unilateral differences become a_i times the original differences. They are equal only when the scaling leaves those differences unchanged, such as a_i = 1 in a nondegenerate case. Conversely, opponent-dependent additions allowed by payoff-difference equivalence may not fit a convention that permits only constants independent of the entire profile.

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How to disprove strategic equivalence

Without relabeling

Find one failed instance of the universal condition. For example, if at a fixed opponents’ profile q player i receives 6 rather than 2 from switching from t_i to s_i in the first game, but 5 in the second, write:

u_i(s_i,q) − u_i(t_i,q) = 6 ≠ 5 = v_i(s_i,q) − v_i(t_i,q).

That single counterexample is sufficient. There is no need to inspect the remaining cells once the stated correspondence has failed.

With relabeling

A failed comparison rules out only the particular player and strategy bijections you proposed. To disprove equivalence when arbitrary relabeling is allowed, either test every permitted correspondence or identify an invariant that every allowed correspondence must preserve and show that the games differ on it. For small games, enumerate the finite permutations explicitly; for larger games, use structural features such as strategy-set cardinalities to eliminate impossible maps before calculating differences.

A practical finite-game checklist

1. Write the objects: list players, pure strategies, profile order, and every payoff.
2. Choose the definition: quote the payoff-difference identity or state the exact transformation used by your source.
3. Fix correspondence maps: identify players and strategies, or state that labels are already identical.
4. Choose a comparison order: for each player, pair of own strategies, and fixed opponents’ profile, subtract the payoffs.
5. Audit every equality: use a table for a small matrix and algebra for a parameterized game.
6. Stop on a counterexample: record the player, fixed opponents, own-strategy pair, and two unequal differences.
7. State the consequence: say whether best responses and Nash equilibria are preserved under the correspondence, and separately note whether payoff levels are preserved.

Common mistakes and how to fix them

Comparing cells instead of switches

Equal entries at corresponding profiles are stronger than necessary, while unequal entries do not disprove difference equivalence. Always subtract two own-strategy payoffs at the same opponents’ profile.

Changing the opponents while comparing

The two terms in a unilateral difference must use one identical opponents’ profile. Comparing a row payoff in one column with a payoff in another column tests a different question.

Checking only best responses

Matching best responses shows a ranking pattern, not the size of each incentive. Two games can agree on which action is optimal while assigning different payoff gains to that action.

Assuming the same equilibria prove equivalence

Equilibrium is an outcome condition. It does not encode every off-equilibrium unilateral comparison, so an identical equilibrium set is insufficient evidence for strategic equivalence.

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Silently switching definitions

A proof that starts with payoff differences and ends by citing a positive affine transformation has not established one coherent relation. Name the convention in the first paragraph and use it consistently.

Ignoring degeneracy

If a player’s payoff difference is zero for a particular comparison, a scaling factor other than one can leave that one equality unchanged. Check all comparisons before drawing conclusions about a transformation; do not infer a global relation from one zero difference.

What is preserved—and what is not

Under payoff-difference equivalence, for every fixed opponents’ profile, each player has exactly the same ordering and tie structure over their own strategies. Therefore the best-response correspondence is the same after the profile identification. Any Nash equilibrium, including a mixed equilibrium described by indifference and best-response conditions, maps to a Nash equilibrium in the other game.

The relation does not generally preserve total welfare, Pareto comparisons across players, sums of payoffs, side payments, or the interpretation of payoff units. Adding a term that depends on opponents’ actions can change the ranking of entire profiles by social surplus even though it leaves every unilateral incentive unchanged. If an application needs cardinal payoff comparisons or welfare conclusions, use a stronger payoff-preservation requirement and prove that requirement directly.

When an algebraic proof is faster than a table

For parameterized games, write the second payoff function as a candidate transformation and simplify the difference:

v_i(s_i,s_-i) − v_i(t_i,s_-i)

Substitute the proposed formula for v_i. If all terms depending only on s_−i cancel and the expression reduces identically to u_i(s_i,s_-i) − u_i(t_i,s_-i), the equality holds for every parameter value covered by the formula. If a remaining term depends on s_i, t_i, or the comparison itself, the proposed equivalence fails unless the stated parameter restrictions force that term to zero.

For a finite but large game, the same logic can be implemented as an exhaustive checklist: iterate over players, opponents’ profiles, and own-strategy pairs; compute both differences; and report the first mismatch with its coordinates. The mathematical proof is still the finite set of equalities, while the program is only a way to avoid transcription errors.

Final proof template

You can adapt the following paragraph to a written solution:

“We use payoff-difference strategic equivalence. The games have the following player and strategy correspondence: … For each player i, each pair of own strategies s_i, t_i, and each fixed opponents’ profile s_−i, we compute u_i(s_i,s_−i)−u_i(t_i,s_−i) and the corresponding difference in the second game. The resulting expressions are equal in every case [or: at the listed case, the values are …]. Therefore the games are [or are not] strategically equivalent under this correspondence. Under equivalence, best responses and Nash equilibria are preserved, although the payoff levels need not be.”

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