How to Prove Two Finite Games Are Strategically Equivalent

To prove two finite normal-form games are strategically equivalent, first state which definition you are using. Under the payoff-difference definition, keep opponents’ strategies fixed, subtract the payoffs from every pair of a player’s own strategies, and show that every resulting difference is identical in the two games. If player or strategy labels differ, specify bijections before comparing profiles. This finite checklist proves the relation; one unequal difference disproves it.

Define “strategically equivalent” before doing any algebra

Game theory uses strategically equivalent for more than one relation. A proof is incomplete if it moves between these meanings without warning. The most direct convention for finite games compares unilateral payoff changes: a player’s payoff difference between two own strategies must be the same at every fixed profile of the other players. Other treatments use positive affine payoff transformations, allow players or actions to be renamed, or require a stronger payoff-preserving isomorphism.

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This article uses payoff-difference equivalence as its main test. The games initially have the same players and the same pure-strategy labels. Later sections show how to adapt the proof when bijective relabeling or another convention is intended.

Matching equilibria or a best-response picture alone is therefore not a proof of payoff-difference equivalence. Those are weaker outcome or preference-pattern comparisons unless the definition explicitly says otherwise.

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Formal setup for a finite normal-form game

Write a game as G=(N,(S_i)_{i∈N},(u_i)_{i∈N}). Here N is the finite player set, S_i is player i’s finite pure-strategy set, and u_i:S_1×⋯×S_n→ℝ is that player’s payoff function. A pure-strategy profile is s=(s_i,s_-i), where s_-i collects the strategies of everyone other than i.

Let H have payoff functions v_i on the same sets. Under payoff-difference strategic equivalence, require

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

for every player i, every two own strategies s_i,t_i∈S_i, and every opponents’ profile s_-i∈S_-i. The opponents’ profile is held fixed in each comparison. This is a finite family of equalities, so it can be checked exhaustively.

The equality is about changes caused by a unilateral switch, not about the absolute numbers in a payoff table. It also preserves ties: if two own strategies give equal payoffs in one game at a fixed opponents’ profile, they give equal payoffs in the other.

A proof procedure that is hard to misapply

1. List both games completely

Write down the player set, each pure-strategy set, and every payoff function. Do not infer that two games have the same domain merely because their tables have the same dimensions. A three-by-three table can represent different strategy labels, different player roles, or different payoff ordering.

If a game has a different number of players or a different number of strategies for a corresponding player, it cannot satisfy a same-label comparison. If the intended definition permits renaming, record that explicitly instead of silently treating labels as interchangeable.

2. State the permitted correspondence

For same-label equivalence, identify player i with player i and each strategy with the identically named strategy. If labels differ, give a player bijection φ:N_G→N_H and, for each player, a strategy bijection ψ_i:S_i^G→S_{φ(i)}^H. The profile map sends s=(s_i)_i to Ψ(s)=(ψ_i(s_i))_i, with the player positions rearranged by φ.

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Every later comparison must use this same correspondence. A proof that compares one row under one labeling and another row under a different labeling has not tested a single mathematical relation.

3. Hold opponents fixed and enumerate own-strategy pairs

For each player, choose every unordered pair of distinct own strategies. For each opponents’ profile, compute the two payoffs being subtracted. If player i has m_i strategies and the opponents have M_-i=∏_{j≠i}m_j pure profiles, there are binom(m_i,2)M_-i pair comparisons for that player. Summing over players gives the total number of checks.

You may use ordered pairs instead; then each comparison appears twice with the sign reversed. Unordered pairs are shorter and are sufficient.

4. Compare the differences, not just their signs

For each row of your calculation, evaluate the difference in G and the corresponding difference in H. Under this definition, both numbers must match exactly. Merely having the same sign proves preservation of preference at that comparison, but it does not prove equality of payoff changes.

5. Record a transparent table or algebraic identity

A small game is easiest to audit in a table with columns for the player, opponents’ profile, own-strategy pair, difference in G, and difference in H. For parametrized games, an algebraic identity is clearer: simplify the two differences and show that their expressions are identical for every allowed parameter and opponents’ profile.

6. Conclude at the exact strength tested

If every required equality holds, the games are strategically equivalent under the stated payoff-difference convention and correspondence. The conclusion does not say that the payoff tables are identical. If even one equality fails, the proposed equivalence is false.

Worked example with two 2-by-2 games

Consider two players. Player 1 chooses U or D; player 2 chooses L or R. Payoffs are listed as (u_1,u_2).

Call this game G. Now consider H:

For player 1, hold player 2’s strategy fixed. At L, switching from D to U changes the payoff by 3−1=2 in G, and by 8−6=2 in H. At R, the corresponding changes are 0−2=−2 and (−2)−0=−2.

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For player 2, hold player 1’s strategy fixed. At U, switching from R to L changes the payoff by 2−0=2 in G, and by 9−7=2 in H. At D, the changes are 1−3=−2 and (−3)−(−1)=−2.

These are all the required comparisons: each player has one strategy pair and there are two opponents’ profiles for each player. Therefore G and H are payoff-difference strategically equivalent under the identity labeling, despite having different payoff levels.

The algebraic shortcut: opponent-only payoff additions

For games on the same labeled strategy sets, the exhaustive condition has an especially useful equivalent form. Define d_i(s_i,s_-i)=v_i(s_i,s_-i)−u_i(s_i,s_-i). If all unilateral differences agree, then for any two own strategies s_i and t_i,

d_i(s_i,s_-i)−d_i(t_i,s_-i)=0.

Thus, for a fixed opponents’ profile, d_i has the same value for every own strategy. There is a function k_i of the opponents’ profile alone such that

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

for every profile. Conversely, any payoff adjustment of this form cancels when two own strategies are subtracted, so it guarantees payoff-difference equivalence.

This characterization is often the fastest proof. Compute v_i−u_i and show that the result is independent of s_i. In the example, player 1 receives an added amount of 5 when player 2 chooses L and −2 when player 2 chooses R. Player 2 receives an added amount of 7 when player 1 chooses U and −4 when player 1 chooses D. Each addition depends only on the opponent’s choice.

The condition is player-specific: a term that depends on player i’s own strategy generally does not cancel and must be tested rather than dismissed as a harmless constant.

What changes when payoffs are transformed or labels are renamed?

Positive affine transformations are not the same test

A common alternative writes v_i=a_i u_i+b_i, where a_i>0. Then a unilateral difference in H is a_i times the corresponding difference in G. Positive scaling preserves which own strategy is better, and therefore preserves best responses, but it does not generally preserve the numerical difference required above. Exact payoff-difference equivalence requires a_i=1 unless every tested difference for that player is zero.

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The additive constant b_i cancels in differences. A more general opponent-dependent addition k_i(s_-i) also cancels, as shown in the preceding section. Do not assume that a source’s use of “affine equivalence” silently means the exact-difference convention.

Strategic isomorphism uses bijections

A structural comparison can allow a permutation of players and a renaming of each player’s strategies. Under an exact-payoff version, the correspondence must satisfy

v_{φ(i)}(Ψ(s))=u_i(s)

for every profile s. This is stronger than payoff-difference equivalence: it preserves the actual payoff at every matched profile. A version that preserves differences instead replaces exact equality with the corresponding quantified difference identity after applying φ and the ψ_i maps.

When labels differ, first prove that the proposed maps are bijections. Then make a profile-by-profile correspondence table or state the map symbolically. Only after that should you subtract payoffs.

Response-graph matching has a narrower claim

A response graph records which unilateral moves improve, worsen, or leave a payoff unchanged. An isomorphism of those graphs can establish that the same response pattern appears after relabeling. It does not establish equality of the magnitudes of payoff changes. A graph edge labeled “improving” is compatible with a gain of 1 in one game and 100 in another.

Coincident equilibria are not enough

Two games can have the same Nash equilibrium set while differing at off-equilibrium profiles or in the size of unilateral gains. Equilibrium coincidence is therefore an outcome observation, not a proof of the universal difference condition. To prove strategic equivalence, still check every player, opponents’ profile, and own-strategy pair required by the definition.

Why best responses and Nash equilibria are preserved

Fix player i and an opponents’ profile s_-i. For any two own strategies, the equality of differences implies

u_i(s_i,s_-i)≥u_i(t_i,s_-i) if and only if v_i(s_i,s_-i)≥v_i(t_i,s_-i).

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All pairwise comparisons, including ties, therefore agree. The set of best responses to s_-i is the same in both games under the specified identification. A profile is a pure Nash equilibrium exactly when every player’s chosen strategy is a best response to the others, so corresponding profiles are equilibria in one game if and only if they are equilibria in the other.

The same reasoning extends to mixed opponents. Expected payoff differences are weighted sums of the pure-profile differences, and every term in those sums is equal across the games. Hence mixed best-response sets and mixed Nash equilibria are also preserved under the natural multilinear extension, provided the same correspondence is used.

How to disprove equivalence efficiently

Use one failed comparison

To refute a proposed same-label equivalence, it is enough to find one player, one fixed opponents’ profile, and one pair of that player’s strategies for which the differences are unequal. For example, if player 1’s change from U to D at L is 2 in one game and 3 in the other, the universal condition fails immediately. You do not need to inspect the remaining cells once that witness is stated.

Distinguish a failed map from a failed isomorphism claim

If relabeling is allowed and one proposed correspondence produces a mismatch, that rejects only that correspondence. A complete disproof must test every permitted correspondence or use an invariant that no allowed relabeling can change. Useful preliminary invariants include the number of players, the multiset of strategy-set sizes attached to players, and the pattern of equality and strict preference comparisons. If those already differ, no bijection can work. If they do not, perform the full mapped difference test.

Finite verification as an algorithm

For identical labels, the following pseudocode implements the definition directly:

for each player i:
    for each opponents profile s_minus_i:
        for each unordered pair (a, b) in S_i:
            delta_G = u_i(a, s_minus_i) - u_i(b, s_minus_i)
            delta_H = v_i(a, s_minus_i) - v_i(b, s_minus_i)
            if delta_G != delta_H:
                return not equivalent
return equivalent

For a relabeled comparison, replace each strategy and player in the second game by its image under the declared bijections before calculating delta_H. With integer or rational payoffs, use exact arithmetic. With floating-point data, define a numerical tolerance in advance and report it; an approximate numerical match is not the same as an exact mathematical proof.

The opponent-only shortcut can reduce work: calculate v_i−u_i for every cell and check that it is constant across each set of cells sharing the same opponents’ profile. This is equivalent to the pairwise test for same-label games, but the pairwise table remains the clearest certificate when the game is small.

Common mistakes and their fixes

• Comparing raw payoff tables cell by cell. Different numbers do not disprove payoff-difference equivalence. Subtract the two own-strategy payoffs at a fixed opponents’ profile.
• Checking only the sign of each difference. Equal preference ordering is weaker than equal payoff change. Compare the numerical differences themselves.
• Letting the opponents vary during a subtraction. A unilateral difference always holds s_-i fixed. Subtracting payoffs from different opponent profiles tests another property.
• Calling every additive term harmless. Only an addition independent of the acting player’s own strategy cancels. A term depending on that strategy must be included in the difference.
• Treating positive scaling as exact equality. Multiplication by a positive number preserves rankings but generally changes the differences.
• Relabeling without declaring the map. State the player and strategy bijections once and use them for every profile.
• Checking only equilibria or a response graph. Those comparisons can miss unequal off-equilibrium payoff changes.
• Forgetting ties. Equality of differences preserves indifference as well as strict preference; a proof that records only winners loses information.
• Using a stronger conclusion than the evidence supports. A difference proof establishes the stated strategic relation and its response consequences, not necessarily exact payoff preservation or a particular transformation convention.

A compact proof template

You can present a finished proof in the following order:

1. Define the two finite games and state whether players and strategies share labels.
2. State the exact meaning of strategic equivalence being used.
3. If needed, specify the player and strategy bijections.
4. For arbitrary i, s_i, t_i, and s_-i, write the two unilateral differences.
5. Show algebraically that the differences are equal, or list every finite comparison in a table.
6. Conclude equivalence under that definition and correspondence.
7. State the resulting preservation of best responses and Nash equilibria, without claiming that payoff levels are unchanged unless you proved that stronger property.

This format makes the quantifiers visible, separates relabeling from payoff comparison, and gives a reader a finite certificate that can be checked independently.

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