iTechGuides is reader-supported. When you buy through links on our site, we may earn an affiliate commission. As an Amazon Associate I earn from qualifying purchases. Learn more
Two games are strategically equivalent only relative to a stated comparison rule. For the standard expanded strategic-form test, each player’s strategy set must have a one-to-one, onto correspondence with the other game’s strategies, and every corresponding strategy profile must produce the same outcome and payoff association. Renaming strategies is therefore necessary only when the renaming is a complete bijection and preserves the entire payoff structure—not merely the number of strategies or one equilibrium.
If move order, information sets, observations, or other sequential features matter, a normal-form match is insufficient. Strategic form can omit those features, so compare the extensive-form trees or use a structural equivalence notion suited to the question.
| # | Preview | Product | Price | |
|---|---|---|---|---|
| 1 |
|
Game Theory 101: The Complete Textbook | $14.99 | Buy on Amazon |
| 2 |
|
Game Theory: An Introduction | $48.43 | Buy on Amazon |
| 3 |
|
Game Theory | $65.57 | Buy on Amazon |
| 4 |
|
A Course in Game Theory (Mit Press) | $55.99 | Buy on Amazon |
| 5 |
|
Game Theory: 100+ Exercises in Strategy and Decision with Detailed Solutions to Master Nash... | $25.99 | Buy on Amazon |
What “strategically equivalent” means
“Equivalent” is not a single universal property in game theory. You must first identify the game representation and the information you want to preserve. A strategic (or normal-form) game usually specifies players, each player’s strategy set, possible outcomes, and an outcome function, with preferences or utility functions attached to outcomes. The Stanford Encyclopedia of Philosophy describes this separation between the game form and the players’ preferences in its discussion of normal-form games: strategic game forms, outcomes, strategies and utilities.
Recommended Free Tools
For the precise expanded-strategic-form criterion set out by Alvin E. Roth and J. Keith Murnighan, two games are strategically equivalent when, for every player, there is a transformation from that player’s strategies in the first game to that player’s strategies in the second game that is both one-to-one and onto, and the outcome/payoff association agrees for every strategy profile after the transformation. Their formal definition appears in Sociological vs. Strategic Factors in Bargaining.
#1 Best Overall
In practical terms, you are asking whether the second game is the first game with strategy labels changed, provided that the relabeling preserves what every complete combination of choices does. If even one mapped profile has a different outcome or payoff under the criterion you selected, that proposed correspondence does not establish equivalence.
Choose the property you intend to preserve
Write the target of the comparison before looking for a mapping. A game can match another on one dimension and differ on another. For example, two matrices may encode the same strategic incentives while representing different timing or information structures.
| Axis | What to inspect | Why it matters |
|---|---|---|
| Players | The same players, or an explicitly justified correspondence between players | Player identity determines whose strategies and preferences are being compared. |
| Strategy sets | A one-to-one and onto mapping for each player | Bijections are part of the cited strategic-equivalence definition; equal counts alone are not enough. |
| Profile mapping | Every joint strategy profile after applying the individual mappings | The criterion quantifies over all profiles, not only equilibria or selected examples. |
| Outcomes and payoffs | The outcome function and each player’s payoff association | These must agree under the selected strategic-form definition. |
| Preference interpretation | Exact utility values or only preference rankings | Utility numbers can represent preferences, but different equivalence notions permit different numerical transformations. |
| Timing and information | Move order, information sets, observations and transitions | Normal form may omit sequential and informational properties that matter strategically. |
| Intended conclusion | The equilibrium, prediction or solution property you want to transfer | Only conclusions invariant under your stated equivalence are licensed. |
The Stanford Encyclopedia’s overview of game theory explains why this distinction matters: normal-form games are commonly represented by matrices, while extensive-form games are represented by trees that record sequences and players’ information. See Game Theory (Summer 2010 Edition).
A step-by-step test for strategic equivalence
1. Identify the representation
Label each object explicitly as a strategic-form game, an extensive-form game, or another model. If you have a matrix, list the players, row and column (or higher-dimensional) strategy sets, outcomes and payoffs. If you have a game tree, record the nodes, actions, information sets, terminal outcomes and player preferences before reducing anything to a matrix.
Do not call two extensive descriptions equivalent merely because their normal-form matrices look alike. A matrix records complete contingent strategies, but it does not by itself show when actions occur or what a player knew when choosing.
2. State the equivalence target in one sentence
Use a sentence such as: “I require equality of outcome and payoff associations after strategy relabeling,” or “I require only preservation of each player’s preference ordering,” or “I require the extensive trees to preserve move order and information sets.” This prevents an unnoticed switch between cardinal and ordinal criteria.
The Roth–Murnighan definition is a precise test for expanded strategic form, not a universal definition for every game-theoretic comparison. Structural notions such as isomorphism and bisimulation are discussed for broader game models in Stanford’s Game Equivalence: Handout.
Crashes, No Sound, or Screen Glitches?
Random freezes, missing sound and display glitches usually trace back to one bad driver. Find and replace yours safely.Free scan · under a minutePC Slower Than It Used to Be?
A free scan shows the junk files, broken settings and background clutter dragging Windows down - then fixes them in one click.Free scan · Windows 10 & 11Rank #2
3. Match the players
Decide whether player 1 in Game A corresponds to player 1 in Game B, or whether a permitted player permutation is part of your comparison. State the correspondence instead of assuming it. A player swap can change whose payoff is being checked, so it should be treated as an explicit transformation rather than an informal relabeling.
4. Propose a bijection for every strategy set
For each player i, write a function from that player’s strategies in Game A to the corresponding strategies in Game B. It must be:
- One-to-one: no two strategies in Game A map to the same strategy in Game B.
- Onto: every strategy in Game B is reached by the mapping.
- Defined for the complete set: no strategy can be left out because it is unattractive, dominated or absent from an equilibrium.
For finite sets, equal cardinality is necessary for a bijection. It is not sufficient: two games can both have two strategies for each player while their payoff patterns fail to match.
5. Map every profile
Take a complete profile from Game A, apply each player’s strategy map, and inspect the corresponding profile in Game B. If Game A has profiles (s1, s2) , the candidate map produces (T1(s1), T2(s2)) . Repeat for every combination, including profiles that are never reached in an equilibrium.
Quick wins for a faster PC:
Fix the driver behind crashes, sound loss and screen glitchesFind Drivers →Clear out junk files and repair common Windows errorsFree Scan →Compare the outcome and payoff association under your stated criterion. One mismatch disproves that particular mapping. If strategy labels are arbitrary, test alternative bijections rather than concluding immediately that the games are nonequivalent.
6. Check the payoff or preference interpretation
Decide what “agree” means numerically. If the definition requires preservation of payoff associations, compare the relevant payoff values profile by profile. If the intended claim is ordinal, compare the induced preference rankings over outcomes instead.
Do not assume that every positive affine or monotone rescaling is automatically allowed. Utility functions can represent the same preference relation while using different numbers, but a definition requiring equality of the payoff association may reject that rescaling. Keep the numerical criterion and the preference criterion separate in your report.
Rank #3
7. Test sequential structure separately when needed
For an extensive game, compare the tree, the order in which players move, each information set, observations available at a decision, and any transition structure relevant to the claim. A normal-form correspondence may show that the same complete strategic possibilities are available while failing to show that players face them at the same time or with the same information.
What’s actually slowing this PC down?
Pick the symptom - the matching free tool is one click away.
A small worked example: relabeling that succeeds
Consider two two-player matrix games. In Game A, the row player chooses Left or Right, and the column player chooses Up or Down:
| Up | Down | |
|---|---|---|
| Left | (3, 2) | (0, 0) |
| Right | (1, 1) | (2, 4) |
Game B uses the labels A, B for the row player and X, Y for the column player:
| X | Y | |
|---|---|---|
| A | (2, 4) | (1, 1) |
| B | (0, 0) | (3, 2) |
Try the bijections Left → B, Right → A, Up → Y, and Down → X. The profile (Left, Up) maps to (B, Y), which has payoff (3, 2), matching Game A. The other three profiles also match under the same maps:
| Game A profile | Mapped Game B profile | Payoff check |
|---|---|---|
| (Left, Up) | (B, Y) | (3, 2) = (3, 2) |
| (Left, Down) | (B, X) | (0, 0) = (0, 0) |
| (Right, Up) | (A, Y) | (1, 1) = (1, 1) |
| (Right, Down) | (A, X) | (2, 4) = (2, 4) |
Under the exact payoff-preservation criterion, this mapping establishes strategic equivalence for these two strategic-form games. The labels themselves carry no strategic content in this comparison.
A near miss: equal strategy counts are not enough
Suppose Game C has the same two-by-two dimensions as Game A, but its upper-left cell is (3, 1) rather than (3, 2), with every other cell unchanged. You can still create bijections between the strategy sets because both games have two strategies per player. However, every candidate mapping must place some Game C profile against the changed payoff cell. Under exact payoff preservation, that profile fails the test, so the games are not strategically equivalent under that mapping.
The conclusion could differ under a deliberately ordinal criterion if the altered numbers leave every player’s preference ordering unchanged. That would be a different claim and must be reported as such; it does not turn the failed cardinal test into a successful one.
Rank #4
Payoffs, utilities and preferences: avoid a category error
A payoff number can be used as a utility representation of a player’s preferences. The representation may preserve only order: an outcome assigned 10 can be preferred to one assigned 5 even if another utility scale assigns them 100 and 1. The relevant question is whether your equivalence notion compares the numbers themselves, the outcome-to-payoff association, or only the rankings that those numbers represent.
- Exact payoff equivalence: corresponding profiles must have the required identical payoff association after relabeling.
- Ordinal preference equivalence: each player must rank corresponding outcomes in the same way, with numerical rescaling allowed only if the stated framework permits it.
- Outcome equivalence: corresponding profiles must produce the same outcomes, while preferences may be analyzed separately.
Write the chosen standard beside your result. “Equivalent” without this qualifier is too broad for a reproducible comparison.
The Tool Desk
Outbyte Driver Updater FREEFix the driver behind crashes, sound loss and screen glitchesFind Drivers →Outbyte PC Repair FREEClear out junk files and repair common Windows errorsFree Scan →Why matching normal forms may not preserve a sequential game
Normal form lists complete strategies, including contingent instructions for actions that may never be reached. It can therefore encode strategic possibilities while hiding the path by which those choices arise. Extensive form displays the sequence of moves and the information available at each decision node. The Stanford Encyclopedia explains that the extensive form contains information about sequences of play and players’ levels of information that the normal form can omit: Game Theory (Summer 2010 Edition).
Use a richer comparison whenever the question involves:
- Who moves first, second or later.
- Whether a player observes an earlier action.
- Which decision nodes belong to the same information set.
- Whether an action is available after a particular history.
- Subgame structure, information-dependent behavior or sequential rationality.
- Transitions or observations in a dynamic model.
Two trees can yield the same strategic form yet differ in these properties. Conversely, a transformation that changes contiguous move order may preserve a particular set-theoretic form while not preserving every semantic feature. The Stanford game-equivalence handout presents structural ideas including isomorphism and bisimulation and discusses transformations of extensive games; treat those as model-specific tools, not a universal replacement for the profile-by-profile test.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How to handle larger games
For a finite game, the comparison can be performed systematically rather than by visual inspection. A practical workflow is:
Do these 3 things before closing this tab:
1Clear out junk files and repair common Windows errors2Fix the driver behind crashes, sound loss and screen glitches3Repair Windows errors before they cause bigger problems- Enumerate each player’s strategies in both games.
- Generate candidate bijections for each player, ruling out any pair with unequal set sizes.
- For each combination of candidate maps, enumerate every joint strategy profile in Game A.
- Apply the maps to obtain the corresponding profile in Game B.
- Compare outcomes and payoffs under the selected exact or ordinal rule.
- Record a mapping only if all profiles pass.
- If no mapping passes, report that no strategic equivalence was found under that criterion and explain which profiles fail for the closest candidates.
For large finite sets, pruning helps: a mismatch in one profile eliminates the current combination of maps, while a distinctive payoff pattern can restrict which strategies could correspond. These are efficiency techniques, not changes to the definition; the final claim still depends on all profiles being checked.
Common mistakes and how to correct them
Comparing only equilibria
An equilibrium is one or a few profiles. Strategic equivalence under the cited definition concerns the entire outcome/payoff mapping. Correct the mistake by checking every profile, including off-equilibrium ones.
Matching strategy counts and stopping
Equal numbers of rows and columns establish only that a bijection might exist. Correct the mistake by writing the actual maps and testing the payoff structure.
Assuming labels determine meaning
“Cooperate” and “Defect,” or “A” and “B,” are labels unless the model assigns an external interpretation to them. Correct the mistake by searching for all plausible bijections rather than matching names literally.
Allowing an unannounced player swap
Switching players can change the payoff comparison. Correct the mistake by stating whether player correspondence is fixed or whether permutations are permitted.
Quietly replacing cardinal equality with ordinal sameness
A monotone transformation may preserve preferences but not exact payoffs. Correct the mistake by naming the criterion before comparing numbers.
Inferring tree equivalence from a matrix
A normal-form match does not establish identical move order or information. Correct the mistake by comparing the extensive structures when those features affect the question.
Claiming every solution concept transfers automatically
Only properties invariant under the selected equivalence are entitled to transfer. Correct the mistake by naming the prediction you are carrying over and checking that its definition depends only on preserved structure.
Free tools Windows power users keep installed
One-click scans. No signup required.
How to report your result clearly
A useful comparison report contains five items:
- Representation: identify whether the games are strategic form, extensive form or another model.
- Criterion: state exact payoff association, outcome preservation, preference ordering or a structural condition.
- Player correspondence: list any fixed identities or permitted player permutation.
- Strategy maps: give the one-to-one, onto mapping for every player.
- Verification: show that every profile passes, or list the first failing profiles and explain the mismatch.
Then add a separate statement about sequential structure if the games have trees. For example: “The strategic forms are equivalent under the stated payoff-preserving relabeling, but this result does not assert equivalence of move order or information sets.” That wording prevents a valid narrow result from being read as a broader claim.
Quick Recap
Decision checklist
- Have you named the game representation?
- Have you stated exactly what must be preserved?
- Are player correspondences explicit?
- Is every strategy map one-to-one and onto?
- Have you checked every joint profile rather than selected outcomes?
- Have you separated exact payoff equality from preference-order preservation?
- Have you tested alternative strategy bijections where labels are arbitrary?
- Have you checked move order, information sets and observations when the game is sequential?
- Does your final conclusion claim only the equivalence that your test establishes?
Product prices and availability are accurate as of the date/time indicated and are subject to change. Any price and availability information displayed on Amazon at the time of purchase will apply.
