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A game is unique only relative to the features and comparison rule you choose. Two games may look different yet generate the same strategic choices and payoffs. Conversely, games with the same normal-form table can differ in move order, information, or credible behavior. To compare games accurately, first choose a representation—strategic (normal) form or extensive form—then state exactly which structure must be preserved.
Uniqueness is model-relative
In ordinary language, a game is a rule system: players make legal choices, the rules determine how play proceeds, and the game ends with an outcome. In game theory, that description becomes a model. A model records selected features of play and ignores others. The features retained determine whether two games count as the same.
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For example, a model might retain only each player’s complete strategy set and payoff for every strategy profile. Under that comparison, a sequential game and a simultaneous game can be equivalent if they produce the same reduced normal form. A model that records the game tree will distinguish them because the order of moves and the information available at each decision differ.
There is therefore no universal answer to “Are these games the same?” A defensible answer has the form: “They are equivalent under relation R because features F, G and H are preserved.” If the relation changes, the verdict can change too.
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What a formal game model records
An extensive-form description normally specifies the following elements. Together they turn prose rules into a mathematical object that can be compared with another object.
- Players: the people, teams, firms, automated agents or other decision makers.
- Positions or nodes: the situations that can occur during play.
- Choices: the legal actions available at each nonterminal position.
- Move ownership: which player acts at each position, or whether a chance event occurs.
- Move order: the sequence in which decisions and chance events happen.
- Information: what a player can observe when choosing. In imperfect-information games, information sets group decision points that the player cannot distinguish.
- Chance: random events and, when modeled, the probability assigned to each possible result.
- Termination: the conditions that stop play.
- Terminal outcomes: what happens at each ending position, such as a win, loss, draw, score or allocation.
- Payoffs: numerical preferences over outcomes when strategic incentives or equilibrium predictions are being analyzed.
Not every model needs every field. A puzzle may need legal moves and winning conditions but no opponent payoffs. A bargaining model may require utilities and beliefs. State which fields your comparison treats as part of identity instead of assuming that every formalism uses the same definition.
Rules, representations and solution concepts are different things
Ruleset
The ruleset is the underlying instruction system: who may do what, when they may do it, what information they receive, how random events work and how outcomes are determined. Two rulebooks can describe the same ruleset with different wording, diagrams or notation.
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A representation is a formal encoding of selected rules. A normal-form table records strategy profiles and payoffs. An extensive-form tree records positions, branches, information sets and terminal results. Translation into a representation can discard distinctions that the original rules contain.
Solution concept
A solution concept is a rule for evaluating or predicting behavior, such as Nash equilibrium, subgame-perfect equilibrium or a refinement that responds to information sets. A solution concept may inspect details that another representation omits. Thus, two games can match under one representation while producing different conclusions under a tree-sensitive solution concept.
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Strategic (normal) form: compare strategies and payoffs
A strategic-form, also called normal-form, game lists the players, each player’s strategy set and a payoff function for every combination of strategies. A strategy is a complete contingent plan: it specifies what a player would do wherever that player might be called upon to act, including situations that are never reached.
| Question | What strategic form shows | What it may omit |
|---|---|---|
| Who participates? | Player set | When a player acts |
| What can each player choose? | Complete strategy sets | The separate decisions that make up a strategy |
| What results from choices? | Payoff for each strategy profile | The path and timing that generated the result |
| How is uncertainty represented? | It can include strategies or expected payoffs | Which information was available at each decision, unless encoded separately |
This form is useful when the question concerns incentives over complete plans and the order of play is irrelevant to the analysis. It is also compact for comparing payoff structures. The caution is fundamental: converting an extensive-form game to strategic form is generally many-to-one. Different trees can produce the same strategy sets and payoff table.
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An extensive-form game is represented as a tree (or an equivalent directed structure). The root is the starting position. Each edge is a legal action or chance result, and each terminal node carries an outcome or payoff. Every nonterminal node identifies the player who moves or a chance mechanism. Information sets show which nodes a player cannot tell apart when acting.
This representation answers questions that a payoff table cannot answer by itself: Who moves first? Can the second player observe the first move? Is a threat made at an unreached node credible? Does a random event occur before or after a decision? Are two decision points distinguishable to the player making the choice?
“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.
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For games with simultaneous moves, the tree uses information sets to show that a player chooses without observing another player’s contemporaneous action. For games with hidden information, several possible positions may belong to one information set. Omitting those sets can accidentally give a player knowledge the rules do not provide.
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What “strategically equivalent” should mean
The phrase is not self-defining. A common criterion in MIT 14.126 lecture notes (Spring 2024) is that two games have the same reduced normal-form representation. Reduced normal form removes strategically redundant strategies while preserving the relevant payoff structure under that reduction.
Under this criterion, two extensive-form games may be strategically equivalent even when their trees differ. The criterion answers a narrow question: do the games induce the same reduced strategic interaction? It does not assert that the rules, information, timing or every equilibrium notion are identical.
Tree-based equilibrium concepts can separate games that share a reduced normal form. For instance, a sequential game may have an off-path decision where a player can make a threat. A simultaneous representation records the resulting payoffs but not whether that threat is credible at the unreached node. Subgame-perfect or other extensive-form analyses can therefore yield different interpretations despite strategic equivalence in the reduced table.
Isomorphism: a structure-preserving correspondence
In mathematics, an isomorphism is a one-to-one correspondence that preserves the structure specified by the objects being compared. For game trees, that normally means mapping each relevant node and branch in one representation to a counterpart in the other while preserving the relationships that the chosen model treats as essential.
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There is no single universal definition of “game isomorphism” across all fields. State the preservation requirements explicitly. Depending on the model, an isomorphism may need to preserve:
- the parent-child and reachability structure of nodes;
- which player controls each decision node;
- the labels or identities of legal actions;
- information-set membership and the player’s observations;
- chance nodes and their probability distributions;
- terminal outcomes, scores or payoff vectors;
- the distinction between simultaneous and sequential decisions.
Two trees can be isomorphic after harmless relabeling—such as renaming “left” and “right” branches consistently—while failing to be isomorphic if one tree gives a player an extra observation or changes a chance probability. Whether player names themselves must map identically or may be permuted is another convention to declare.
How to compare two games step by step
- Define the question. Decide whether you care about strategic incentives, predicted equilibria, legal play, information, probability, outcomes or a combination.
- Select the representation. Use strategic form for strategy-set and payoff comparisons. Use extensive form when order, information, chance timing or termination matters.
- Inventory the modeled features. Write down players, actions or complete strategies, move sequence, information sets, chance probabilities, ending conditions, terminal outcomes and payoffs.
- Normalize cosmetic differences. Rename actions or reorder a table only when the relabeling is consistent and has no effect on the modeled structure. Different prose or diagram styles alone do not create a new game.
- Construct a mapping. Match each relevant position, strategy, action and terminal result in one game to an element in the other. Record any unmatched item rather than silently ignoring it.
- Test preservation. Check that the mapping keeps the selected features unchanged: legal choices remain legal, player ownership remains correct, information sets match, probabilities remain the same and corresponding terminal payoffs agree.
- Name the conclusion narrowly. Use wording such as “same payoff matrix,” “same reduced normal form,” “isomorphic under this tree-preserving mapping” or “equivalent for Nash equilibrium.” Do not call games simply “the same” when the relation is narrower.
- Check solution concepts separately. If the practical question is a prediction, state whether the analysis uses Nash equilibrium, subgame-perfect equilibrium or another concept, and verify that the representation retains the information that concept needs.
The feature checklist: what changes make a game different?
| Feature | Diagnostic question | Why a change matters |
|---|---|---|
| Players | Are the decision makers and their roles the same? | Adding, removing or merging a player changes the strategic interaction. |
| Actions | Does every player have the same legal choices at corresponding positions? | An extra action can create new strategies and outcomes. |
| Move order | Who acts before whom? | Order can create commitment advantages and credibility issues. |
| Information | What does each player know at each choice? | Observing an earlier move can change the available contingent plans. |
| Chance | Where do random events occur, and with what probabilities? | Moving a chance event or changing probabilities alters expected results. |
| Termination | When does play stop? | Different stopping rules can permit or forbid entire paths. |
| Outcomes | Do corresponding terminal histories produce the same result? | Different win, draw or allocation rules change what players seek. |
| Payoffs | Are preferences over outcomes identical? | Strategic predictions depend on incentives, not labels alone. |
Worked example: the same table, different trees
Sequential entry and simultaneous entry
Imagine an entrant choosing “Enter” or “Stay out,” followed—only after entry—by an incumbent choosing “Fight” or “Accommodate.” The extensive form records the entrant’s first move, the incumbent’s later move and whether the incumbent observes entry. It can also represent a threat to fight that is never reached.
A strategic-form conversion gives the entrant two strategies and the incumbent complete plans such as “Fight after entry” or “Accommodate after entry.” If another game lets both players choose their strategies without observing any move, it may generate the same reduced payoff table. The table alone cannot tell you whether “Fight” was an observed response, an off-path contingency or a simultaneous choice.
If the question is whether the payoff incentives match after eliminating redundant strategies, the games may be strategically equivalent. If the question is whether the incumbent’s threat is credible, use the extensive form and a tree-sensitive solution concept; the games are not interchangeable for that purpose.
Information-set variation
Suppose two branches of a tree lead to a decision by the same player. In one version the player sees which branch occurred; in the other, both nodes belong to one information set. The legal action labels can be identical, yet the player’s strategy requirements differ. Treating those versions as isomorphic would incorrectly grant information that the second game withholds.
Chance timing variation
Consider a lottery that selects a state before a player chooses versus after the player commits to an action. Even with the same probabilities and terminal payoffs, moving the random node changes what the player knows when acting. A normal-form table may hide that timing; an extensive-form comparison will not.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Cosmetic rule changes versus substantive changes
A rule description can change without changing the modeled game. Renaming “pass” as “skip,” numbering cards differently while preserving all relationships, or drawing a tree from right to left can be cosmetic if the correspondence preserves legal moves, information, probabilities and outcomes.
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A formal game-grammar approach published in 2020 is designed to compare finite, discrete game systems while being insensitive to cosmetic variations in rule descriptions. Its stated scope includes randomness and mixed sequential/simultaneous play but excludes hidden information. Do not extend that method automatically to games outside those conditions.
Common comparison mistakes and how to correct them
- Calling identical payoffs identical games. A payoff table does not retain every dynamic fact. Add the move order and information structure before making a stronger claim.
- Assuming strategic equivalence means behavioral equivalence. State which solution concept is being used. Extensive-form refinements can distinguish games with the same reduced normal form.
- Ignoring information sets. A player who cannot distinguish two nodes must choose consistently across that information set. Drawing separate, fully informed nodes changes the game.
- Dropping chance probabilities. The location and distribution of random events are part of the model whenever expected outcomes matter.
- Comparing rulebook prose instead of structures. Translate both descriptions into the same formal representation, then compare the retained features.
- Using “isomorphic” without a preservation list. Name whether the correspondence preserves labels, player identities, information, probabilities, outcomes and payoffs.
- Confusing a strategy with an action. An action is a move at one position; a strategy specifies actions at every decision point relevant to that player.
- Assuming every formalism handles every game. Check whether the model permits hidden information, continuous actions, infinite play or other features outside its stated domain.
Choosing the right notion for a practical question
| If you want to know… | Use… | Report the result as… |
|---|---|---|
| Whether incentives and payoff rankings match | Strategic form | Same strategic or reduced normal form, if verified |
| Whether two rulebooks describe the same possible play | Extensive form or a rule-level model | Equivalent under a stated ruleset mapping |
| Whether players observe the same events | Extensive form with information sets | Information-preserving isomorphism or a specific difference |
| Whether random outcomes have the same effect | Extensive form with chance probabilities | Chance-preserving equivalence |
| Whether an equilibrium prediction carries over | The representation required by that solution concept | Equivalent with respect to the named solution concept |
When in doubt, report both levels: first say whether the reduced strategic forms match, then identify any extensive-form differences that could affect dynamic or informational analysis.
Further reading and scope notes
The Cambridge Game Theory material presents extensive-form games through their trees, players, choices, information and terminal outcomes, and treats strategic-form games in a separate chapter. The Stanford Encyclopedia of Philosophy entry cited above is the Summer 2012 edition; its warning about order of play applies to analyses in which timing matters. MIT 14.126 lecture notes from Spring 2024 illustrate how representation and solution concepts can diverge, rather than defining one universal equivalence relation. The 2020 game-grammar paper offers a formal comparison method for its specified class of finite, discrete games without hidden information.
A textbook chapter on extensive-form games is useful if you need to construct trees, information sets or equilibrium refinements. It is not necessary for the basic comparison workflow: define the features, choose the representation, test a mapping and state the equivalence claim with its scope.
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