A game is unique only relative to a stated model and comparison rule. Two games can have different-looking boards, stories, or rule text yet be equivalent when they preserve the same choices and payoffs. Conversely, games with the same strategic-form table can differ in move order, information, or chance. To decide whether two games are “the same,” specify which features matter, represent both games at that level, and test whether a structure-preserving mapping keeps those features intact.
Uniqueness is always relative to a model
There is no single, context-free test that decides whether two games are identical. A game can be viewed as a physical object, a written ruleset, a mathematical representation, or a problem paired with a solution concept. Each viewpoint preserves different facts.
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For a casual rules question, two games may be the same if players make the same legal choices and reach the same outcomes. For a strategic analysis, the relevant test may be whether the players have the same strategy sets and payoff functions. For a dynamic question, the order of moves and the information available at each decision are essential. A mapping that is adequate for one purpose can be too coarse for another.
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What a formal game model records
An extensive-form description treats play as a tree. The model can include the following components:
| Component | What it records | Why it can affect identity |
|---|---|---|
| Players | Who makes decisions or receives outcomes | Changing the decision makers changes whose options and incentives are being analyzed. |
| Positions or nodes | The states that can occur during play | Different reachable states can create different continuation choices. |
| Actions or edges | Legal moves from each nonterminal position | Adding, removing, or relabeling a meaningful action can change the game. |
| Move order | Which player or chance event acts at each position | Order determines what a player can condition a decision on. |
| Information sets | Decision points a player cannot distinguish when choosing | They determine what the player knows and which contingent plans are legal. |
| Chance | Random events and their probabilities | Different probabilities alter the distribution of outcomes even when choices look alike. |
| Termination | When play ends and which positions are terminal | Changing ending conditions can create or remove possible continuations. |
| Outcomes | The result assigned to each terminal history | Two games with different result labels may still be equivalent if the labels are merely cosmetic; different consequences are not cosmetic. |
| Payoffs | Numerical or ordinal preferences over outcomes | Payoffs are needed when the question concerns incentives, equilibria, or predicted behavior. |
A Cambridge chapter summary on extensive-form games presents this tree-based structure: positions, the mover or chance at each nonterminal position, available moves, information sets, and terminal outcomes. Payoffs can be attached when strategic incentives are part of the model. Not every analysis needs every field, but a comparison is meaningful only after you state which fields are included.
Keep three layers separate
The ruleset
The ruleset is the instruction system players follow. It determines legal choices, how a move changes the state, when another participant acts, and how play ends. A rulebook can describe the same underlying system in different prose, languages, diagrams, or layouts.
The representation
A representation is a formal encoding of the rules. A game tree makes sequence and information explicit. A strategic-form table lists complete strategies and the resulting payoffs. A compact computer format might encode states and transitions. The representation is not the game itself; it is a selected description that may omit details irrelevant to the current question.
The solution concept
A solution concept is a rule for evaluating or predicting play, such as a Nash-equilibrium test or an equilibrium refinement that examines behavior at every information set. Two representations can support the same basic payoff comparison while yielding different conclusions under a solution concept that inspects the path of play.
Confusing these layers causes many false claims of uniqueness. A rewritten rulebook may preserve the ruleset. A strategic-form conversion may preserve enough information for one payoff question while discarding dynamic facts. An equilibrium result belongs to the model plus the solution concept, not to the title of the game alone.
Strategic form (normal form)
A strategic-form, also called normal-form, game is organized around players’ strategy sets and payoff functions. A strategy is a complete plan for what a player would do in every decision situation that the representation recognizes. The resulting table or function shows the payoff associated with each profile of complete strategies.
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This form is useful when the main question is which plans are available and how their combinations are valued. It is compact for simultaneous-move problems and for analyses where the timing of choices does not matter. It also provides a common language for equilibrium calculations.
Its compression is a limitation. A strategy profile can encode contingent instructions without showing when an instruction would be used, what a player knew at that moment, or which histories are reached. The table therefore does not, by itself, tell you whether one player moved first, whether another observed that move, or whether an unreached branch existed.
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Extensive form (game tree)
An extensive-form game displays the sequence of play. Nodes represent positions, edges represent legal actions, and each nonterminal node identifies the player who acts or a chance event. Information sets group decision nodes that a player cannot distinguish when choosing. Terminal nodes carry outcomes and, when incentives are modeled, payoffs.
This representation is indispensable when order, observation, commitment, hidden information, or random events matter. A player’s strategy must specify an action at every information set where that player might be called upon to move, including branches that equilibrium play may never reach.
“Where order of play is relevant, the extensive form must be specified or your conclusions will be unreliable.”
Stanford Encyclopedia of Philosophy, “Game Theory,” Summer 2012 edition
The statement is a scope warning, not a claim that every comparison requires a tree. If the question concerns only a payoff table, strategic form may be sufficient. If it concerns whether a threat is credible, whether an observation changes behavior, or how a random event is timed, the tree is part of the subject.
Why converting a tree to a table can lose distinctions
Moving from extensive form to strategic form lists complete plans and their payoffs. That conversion can preserve the information needed for some calculations, but it does not generally preserve every dynamic feature. Multiple extensive-form games can induce the same strategic-form representation.
For example, two trees may produce identical sets of complete plans and identical payoff assignments while differing in whether a later player observed an earlier action, whether two decision nodes were joined in one information set, or whether a branch was reached through a different sequence. The strategic table can make those trees look the same because it records the plans and outcomes but not the full path structure.
This matters whenever the analysis uses the tree. An extensive-form equilibrium concept may evaluate behavior at information sets, require sequential rationality, or reject a noncredible continuation. Games sharing a reduced normal form can therefore receive different treatment under an extensive-form solution concept.
What “strategically equivalent” should mean
Strategic equivalence is not a universal phrase. One common criterion, used in lecture material for game theory, is that two games have the same reduced normal-form representation. Under that criterion, redundant strategies are removed and the remaining strategy-payoff structure is compared.
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That relation answers a narrow question: do the games induce the same reduced strategic table? It does not assert that their trees, information sets, chance timing, or unreached continuations are the same. If those features affect the solution concept, a strategic-equivalence claim is insufficient.
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When writing a comparison, state the relation in full:
- Same strategic form: the specified strategy sets and payoff function coincide.
- Same reduced normal form: the comparison also removes the redundant strategies allowed by the chosen reduction.
- Tree-isomorphic: a named mapping preserves the selected nodes, edges, movers, information sets, chance probabilities, and terminal labels or payoffs.
- Equivalent under a solution concept: the chosen concept gives the same relevant prediction or value, even if the underlying representations differ.
Each statement is narrower and more informative than “these are the same game.”
Isomorphism: a structure-preserving correspondence
In mathematics, an isomorphism is a correspondence that preserves the structure under discussion. In game theory, the exact fields included in that structure must be declared; there is no single convention that applies to every game and every research tradition.
For a finite game tree, a proposed isomorphism might map every node in one tree to exactly one node in the other, preserve the parent-child relation, match the player who moves, carry each action to a corresponding action, preserve information-set membership, match chance probabilities, and pair terminal nodes with equivalent outcomes or payoffs. If any of those features is outside the chosen model, it need not be preserved—but omitting it changes the meaning of “isomorphic.”
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Labels versus structure
Renaming players, actions, or terminal outcomes does not necessarily destroy an isomorphism. A relabeling can be harmless when there is a one-to-one translation that preserves the relevant relations and preferences. Calling an action “north” instead of “left” is cosmetic if every rule and payoff association is translated consistently.
When a difference is structural
A difference is structural when it changes a modeled relation: one tree has an extra branch, a player observes a move in one game but not the other, a chance node uses different probabilities, or terminal payoffs are assigned differently. Those changes cannot be dismissed as wording.
Scope matters
A mapping designed for finite, discrete games without hidden information may not apply to games with private information or continuous action spaces. The 2020 game-grammar paper develops equivalence relations for a specified class of finite discrete games, allowing randomness and mixtures of sequential and simultaneous play. Its criterion should be treated as a formalism for that scope, not as a universal definition for every game.
A practical method for comparing two games
Use the following procedure whenever someone asks whether two games are the same, equivalent, or merely similar.
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- State the question. Are you comparing legal play, strategy and payoff structure, dynamic incentives, or the prediction of a particular solution concept? The answer determines the representation you need.
- Choose the representation. Use strategic form for strategy/payoff comparisons when timing is irrelevant to the question. Use extensive form when sequence, information, chance, or termination matters.
- List the recorded features. Write down players, action or strategy sets, move sequence, information at each decision, chance events and probabilities, terminal conditions, outcomes, and payoffs if incentives are being analyzed.
- Construct a mapping. Match the positions and choices of one game to the other. In a tree, check every node and continuation; in a strategic form, check every strategy and payoff entry retained by the comparison.
- Test preservation. Verify that the mapping keeps the selected features unchanged. A bijection of labels is not enough if it changes who knows what or when a move occurs.
- Apply the intended solution concept. If the conclusion concerns equilibrium or predicted behavior, say whether the concept uses only the strategic table or also inspects the extensive form.
- Report a qualified conclusion. Use the narrowest accurate statement: “same reduced normal form,” “isomorphic under the stated tree mapping,” or “equivalent for this solution concept.”
The comparison axes that usually decide the result
| Axis | Questions to ask | Typical consequence of a change |
|---|---|---|
| Players | Are the same participants decision makers, and are their preferences represented the same way? | A new player or a different payoff owner changes the model. |
| Available actions | Does each player have the same legal choices at corresponding positions? | An added or removed meaningful action changes strategy sets. |
| Move order | Who acts first, and can later choices condition on earlier moves? | Changing order can alter incentives even with similar payoffs. |
| Information | Which histories can a player distinguish at a decision point? | Joining or separating information sets changes feasible contingent behavior. |
| Chance | Where do random events occur, and what probabilities do they use? | Different timing or probabilities changes outcome distributions. |
| Termination | Which histories end play, and can cycles continue indefinitely? | New stopping rules create or remove outcomes and strategies. |
| Outcomes | Are terminal results merely renamed, or do they carry different consequences? | Different consequences mean the games are not equivalent under an outcome-sensitive model. |
| Payoffs | Are preferences or numerical rewards preserved, and under what transformations? | Different incentives can change equilibrium and optimal choices. |
Examples of differences that are, and are not, meaningful
Cosmetic presentation changes
Replacing a fantasy theme with a space theme, renaming pieces, or rewriting a rule in clearer language can leave the game unchanged. The test is whether every legal choice, transition, terminal result, and relevant preference has a consistent translation.
A changed action set
If one version lets a player pass and the other does not, the difference is structural even when passing is rarely useful. It adds a strategy and can affect what other players must plan for.
A changed move order
Suppose the same two decisions occur but one version lets the second player observe the first action. The available labels may look identical, yet the second player can now condition behavior on information that was previously unavailable. A strategic-form comparison might conceal this distinction; an extensive-form comparison cannot.
A changed information set
Two decision nodes can have identical actions and payoffs but belong to different information sets. If a player knows which node was reached in one game and cannot tell in the other, the player’s legal contingent plans differ. That is a change in the game, not a typographical variation.
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Replacing a fair random event with one that assigns a 70 percent probability to one branch changes the model, even if the tree shape and action labels remain the same. The probability is part of the structure whenever chance is included.
A changed stopping rule
Allowing a contest to continue until a player reaches three wins instead of ending after one win changes the terminal histories. The games may share local moves but not the same complete set of possible plays.
A changed payoff scale
Whether a payoff change matters depends on the model and the solution concept. If only ordinal preferences are used, a transformation that preserves each player’s ranking may be harmless for that analysis. If numerical expected payoffs, risk attitudes, or mixed strategies are being calculated, the scale and probabilities must be handled explicitly.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Why solution concepts can separate games with the same table
A strategic table can identify equilibria based on complete strategies, but it does not show how a strategy is reached or whether a prescribed action is credible at every information set. Extensive-form concepts can impose additional requirements on off-path behavior and sequential rationality.
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1Clear out junk files and repair common Windows errors2Fix the driver behind crashes, sound loss and screen glitches3Repair Windows errors before they cause bigger problemsConsequently, two games that share a reduced normal form may have different answers to questions such as: Is a threat credible? What should a player do after an unexpected action? Does an observation change the continuation strategy? Which behavior is justified at an unreached node? Those questions require the tree or another representation that retains the relevant dynamic information.
When reporting a prediction, name both the representation and the solution concept. “The games have the same payoff table” is not a complete explanation for a claim about behavior if the claim depends on timing or information.
Using a game-grammar approach carefully
A formal game-grammar framework can compare game systems while ignoring purely cosmetic differences in rule descriptions. Such a framework is useful when you need machine-readable rules or a repeatable equivalence test. Its published scope is finite, discrete games without hidden information, while still allowing randomness and combinations of sequential and simultaneous play.
That scope is a boundary, not a footnote. A result established in that formalism should not automatically be extended to games with private information, continuous actions, unbounded play, or other features the framework does not model. State the class of games and the preserved fields before treating the result as an isomorphism claim.
Common mistakes when declaring games “the same”
- Comparing titles or themes. Names and stories are not formal criteria.
- Using only the payoff table. This can erase order, information, chance timing, and terminal structure.
- Calling every relabeling an isomorphism. Relabeling is harmless only when the translation preserves the selected structure.
- Ignoring unreachable branches. A branch that is never reached under one prediction can still matter to strategies and refinements.
- Leaving “equivalent” undefined. Readers cannot tell whether you mean equal payoffs, equal reduced normal form, tree isomorphism, or equal predictions.
- Mixing rules and analysis. A solution concept is not part of the ruleset; it is a method applied to a chosen model.
- Overgeneralizing a formalism. A criterion developed for a restricted class of finite games may not cover hidden information or continuous play.
A worksheet for a real rule comparison
For a board game, card game, video game mechanic, or abstract puzzle, create two parallel descriptions and fill in these fields:
- List every player, chance source, and decision point.
- Record legal actions at each point and the state transition caused by each action.
- Draw the order of moves, including simultaneous decisions and random events.
- Mark what each player knows at every decision and group indistinguishable positions into information sets.
- Mark all terminal conditions, including draws, cycles, time limits, and resignation rules.
- Translate terminal results into outcomes, then attach payoffs only if the question concerns incentives or preferences.
- Choose the comparison relation and attempt a one-to-one mapping of the recorded structures.
- Test any claimed prediction with the solution concept that the question actually requires.
- Write the conclusion with its scope: identify what is preserved and what was intentionally ignored.
This worksheet prevents a common category error: declaring two games equivalent because their surface rules look alike while silently changing information, timing, or incentives.
Further reading and source scope
The Stanford Encyclopedia of Philosophy entry “Game Theory” (Summer 2012 edition) distinguishes strategic and extensive forms and emphasizes the importance of specifying order when it matters. MIT 14.126 lecture notes from Spring 2024 illustrate how representation and solution concepts can separate games that look equivalent in reduced form. Cambridge’s Game Theory coverage includes separate treatment of extensive-form and strategic-form games. The 2020 game-grammar paper develops a formal equivalence approach for its stated class of finite discrete games without hidden information.
These sources support different parts of the framework. None supplies a single universal definition of game isomorphism, so any article, textbook, or software tool that uses the term should state its own preserved structure.
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Bottom line
What makes a game unique is not its name or appearance but the structure your analysis chooses to preserve. Start with the question, select strategic or extensive form accordingly, compare players, choices, order, information, chance, termination, outcomes, and payoffs, then name the exact equivalence or isomorphism relation. A careful conclusion is usually conditional: two games may be equivalent for one purpose while remaining different for another.
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