Game Form: The Rules of a Game Without the Payoffs

A game form is the set of rules that turns what players do into what happens. It lists the moves each player may make and states, for every combination of moves, which outcome results. What it deliberately leaves out is how anyone feels about those outcomes. A game form is sometimes also called a game frame, a ruleset, or an outcome function. The defining contrast, as the standard reference puts it, is that "a game form differs from a game in that it does not stipulate the utilities or payoffs for each agent."

That sounds like a small technical distinction, but it is the hinge on which a whole field turns. If you separate the rules from the preferences, you can ask a powerful question: given that I do not know what people want, and given that they may lie or act strategically, which rules should I write so that the result is good anyway? That question is the subject of mechanism design and implementation theory, and the game form is its basic building block. Electoral systems, auctions, procurement contests, regulatory contracts, and schemes for paying for shared public projects can all be described as game forms.

This article explains what a game form is, how it differs from a full game, where the idea came from, what the main theorems say, and how the concept is used. Several worked examples show the arithmetic. Hypothetical numbers are labeled as such.

The basic idea in plain terms

Think of a board game sold in a box. The rulebook says what each player can do on a turn and how the game ends: who wins, who pays whom, what happens to the pieces. That rulebook is close to a game form. What the rulebook does not tell you is how much each player cares about winning, whether one player is happy to lose if a friend wins, or whether someone is playing only to finish quickly. Those are the players' preferences. Combine the rulebook with the players' preferences and you get a game in the full game-theoretic sense, one you can analyze for equilibria.

The same separation applies to institutions. A sealed-bid auction is a rulebook: each bidder submits a number, the highest number wins, and the winner pays some amount. That rulebook stays the same whether the item is a painting worth a lot to one bidder or a used car nobody much wants. Only when we add the bidders' valuations do we get a game whose likely play we can predict.

A formal definition

In the usual formalization, there is a finite set of players, numbered 1 through n. Each player i has a set of available actions or strategies, written S_i. There is also a set of possible outcomes, often written X or A. The game form is a function g that assigns an outcome to every profile of actions:

g : S_1 × S_2 × ... × S_n → X

For every list of choices (s_1, s_2, ..., s_n), one per player, g tells you the resulting outcome g(s_1, ..., s_n). Lecture slides on Gibbard's work at the University of Amsterdam state it in exactly this way, as a function from the product of strategy sets to an outcome set. Allan Gibbard's 1973 paper described a game form as any scheme that makes outcomes depend on individuals' actions, which he called strategies.

Nothing in that definition says which outcomes are good. A game form can be analyzed for structural properties, such as whether one player can force any outcome alone, without any reference to preferences. To predict behavior you need to add preferences.

From game form to game

Add to the game form a preference ordering or utility function u_i for each player over the outcomes in X. Now each player's payoff from an action profile is u_i(g(s_1, ..., s_n)). This combination is a game in normal form. The same game form, paired with different preference profiles, produces different games.

Message spaces and direct mechanisms

When the actions players take consist of reporting information, such as their preferences, beliefs or valuations, the action set is called a message space. A game form in which each player's message is simply a report of his or her own private information, often called the player's type, is called a direct mechanism. A sealed-bid auction in which each bidder states a valuation is a direct mechanism. A system in which bidders take turns raising a price out loud is an indirect mechanism, since the messages are not statements of valuation.

The distinction matters because of a theorem known as the revelation principle, discussed below, which says that for many purposes the designer can restrict attention to direct mechanisms in which telling the truth is in each player's interest.

Why separate rules from preferences

The separation exists because the designer of a set of rules usually does not know what the participants want. A government auctioning radio spectrum does not know how much each telecom company values a license. A town choosing whether to build a bridge does not know how much each resident would truly pay for it. A legislature writing an electoral law cannot know how voters will rank candidates in every future election. The rules must be fixed first and must work well across many possible preference profiles.

Writing the rules as a game form makes that challenge precise. The designer picks a game form g. Nature, or the world, then picks a preference profile from some range of possibilities. The players, knowing their own preferences, play the game that results. The designer wants the outcome to be good in some defined sense, for every preference profile in the range, or on average.

Social choice functions: the designer's target

What counts as a good outcome is formalized as a social choice function (or, when several outcomes are acceptable, a social choice rule or correspondence). A social choice function f assigns to each preference profile the outcome the designer would choose if she could see everyone's preferences directly. Examples include choosing the candidate ranked first by the most voters, or giving an item to the bidder who values it most, or building a public project exactly when its total value exceeds its cost.

The designer's problem is that she cannot see preferences. She can only write rules. A game form is said to implement a social choice function when, for every preference profile in the relevant range, the equilibrium outcome of the resulting game is the outcome the social choice function prescribes. When a game form is built to achieve this, it is often called a mechanism or an implementation. The Wikipedia summary of the concept states the designer's task this way: craft the game form so that when each player plays a best response, the resulting equilibrium implements the desired social choice function.

A short history of the idea

The game form as a separate object emerged from several lines of work in the 1960s and 1970s: the economics of information and decentralized resource allocation, the theory of auctions, and the theory of voting. The 2007 Nobel memorial prize in economics recognized the field that grew from it.

Hurwicz and the mechanism as a communication system

Leonid Hurwicz's 1960 paper "Optimality and Informational Efficiency in Resource Allocation Processes," published in a Stanford University Press volume edited by Arrow, Karlin and Suppes, is generally cited as the starting point. According to the scientific background document prepared by the prize committee of the Royal Swedish Academy of Sciences for the 2007 prize, Hurwicz "defined a mechanism as a communication system in which participants send messages to each other and/or to a 'message center,' and where a pre-specified rule assigns an outcome (such as an allocation of goods and services) for every collection of received messages."

That is a game form in all but name. Hurwicz's framework let markets be compared with many alternative institutions on a common footing. The early focus was on informational cost: how large a message space does a mechanism need? A later summary by Mount and Reiter (1974) describes Hurwicz's setup in terms of an "environment," covering goods, agents, preferences, production possibilities and endowments, with a mechanism taking environments to allocations. The question of which mechanism reaches efficient outcomes with the smallest message space was connected to the older debate about whether a centrally planned economy could match the informational efficiency of markets.

Incentive compatibility

The early informational work largely set aside incentives. The Nobel background document says mechanism design "became relevant for a wide variety of applications only after Hurwicz (1972) introduced the key notion of incentive-compatibility, which allows the analysis to incorporate the incentives of self-interested participants." In a direct mechanism, incentive compatibility means it is in each participant's interest to report private information truthfully. Hurwicz also proved a negative result: in a standard exchange economy, under weak assumptions, no incentive-compatible mechanism that respects voluntary participation can always produce Pareto-optimal outcomes. As the committee put it, "private information precludes full efficiency."

Vickrey, Clarke and Groves

Separately, William Vickrey's 1961 paper "Counterspeculation, Auctions, and Competitive Sealed Tenders," in the Journal of Finance, volume 16, analyzed different auction formats and introduced what is now called the second-price sealed-bid auction. Edward Clarke (1971) and Theodore Groves (1973) extended the idea to public goods. The Nobel background document notes that before 1970, economists generally believed public goods could not be provided efficiently because people would not reveal how much they valued them, and that it "came as a surprise" when Clarke and Groves showed that, with quasi-linear utility (no income effects), there is a class of mechanisms in which truthful reporting is a dominant strategy and the chosen level of the public good maximizes social surplus. Because Vickrey's paper already contained the key intuition, these are often called Vickrey-Clarke-Groves (VCG) mechanisms.

Gibbard and the game form proper

The term "game form" in its now-standard sense is closely associated with the philosopher Allan Gibbard. His 1973 paper "Manipulation of Voting Schemes: A General Result," in Econometrica volume 41, pages 587 to 601, studied general game forms with a finite set of outcomes. He called a game form "straightforward" (now usually called strategy-proof) if every player always has a dominant strategy, whatever his or her preferences. His theorem, stated in modern terms, says: if a game form is not dictatorial and has at least three possible outcomes, then it is not strategy-proof.

Mark Satterthwaite proved a closely related result independently in "Strategy-proofness and Arrow's Conditions," Journal of Economic Theory, volume 10 (1975). The combined statement for voting rules based on rankings is called the Gibbard-Satterthwaite theorem. Gibbard's 1973 paper also contained the first version of the revelation principle, for dominant strategies. In 1978 Gibbard extended his analysis to game forms whose outcomes are lotteries, in "Straightforwardness of Game Forms with Lotteries as Outcomes," Econometrica volume 46, pages 595 to 614.

The revelation principle and Bayesian mechanism design

The revelation principle was extended to Bayesian Nash equilibrium, the solution concept for games where players have private information and beliefs about others, by several researchers working independently. The Nobel committee lists Dasgupta, Hammond and Maskin (1979), Harris and Townsend (1981), Holmström (1977), Myerson (1979) and Rosenthal (1978). Roger Myerson's "Incentive Compatibility and the Bargaining Problem," Econometrica volume 47 (1979), is a standard reference, and the committee credits Myerson (1979, 1982, 1986) with developing the principle "in its greatest generality."

Maskin and implementation theory

The revelation principle guarantees that truth-telling can be an equilibrium. It does not guarantee that it is the only one. Eric Maskin's work, first circulated in 1977 and published in 1999 as "Nash Equilibrium and Welfare Optimality" in the Review of Economic Studies, volume 66, pages 23 to 38, gave the first general answer to when a game form can be built so that all of its equilibria are desirable. The abstract states that any social choice rule satisfying two properties, monotonicity and no veto power, "can be implemented by a game form if there are three or more individuals." The Nobel committee notes that Maskin's original manuscript had an incomplete proof and that complete proofs were later provided by Williams (1986), Repullo (1987) and Saijo (1988).

Cooperative game forms and effectivity functions

Another line of work asked what power coalitions have in a game form. Hervé Moulin and Bezalel Peleg's 1982 paper "Cores of Effectivity Functions and Implementation Theory," in the Journal of Mathematical Economics, volume 10, studied effectivity functions, which describe which sets of outcomes each coalition can force. These connect game forms to cooperative game theory and to implementation in strong equilibrium, where no coalition can jointly deviate to its advantage.

The 2007 prize

In 2007 the Royal Swedish Academy of Sciences awarded the memorial prize in economic sciences to Leonid Hurwicz (University of Minnesota), Eric S. Maskin (Institute for Advanced Study, Princeton) and Roger B. Myerson (University of Chicago) "for having laid the foundations of mechanism design theory." The academy said the theory "has helped economists identify efficient trading mechanisms, regulation schemes and voting procedures."

Timeline

How game forms are analyzed

Once you have a game form, analysis proceeds by asking what players with various preferences would do, using a solution concept from game theory. Three solution concepts dominate.

Dominant strategies

A strategy is dominant if it is a player's best choice no matter what the others do. If every player has a dominant strategy that depends only on his or her own preferences, the game form is strategy-proof. This is the strongest and most robust kind of prediction: players need not know anything about one another, need not form beliefs, and need not coordinate. Strategy-proof game forms are therefore highly prized. The second-price auction and the VCG mechanisms are strategy-proof under quasi-linear preferences. The Gibbard-Satterthwaite theorem says that in general voting settings with three or more alternatives, strategy-proofness forces dictatorship.

Nash equilibrium

In a Nash equilibrium, each player's strategy is a best response to the others' strategies. Nash implementation is natural when players know one another's preferences well (complete information) but the designer does not. A legislature, a committee of colleagues, or a small group of firms who know each other's costs might fit this description. Maskin's theorem characterizes which social choice rules can be Nash-implemented.

Bayesian Nash equilibrium

When each player knows only his or her own type and holds beliefs (a probability distribution) about others' types, the relevant concept is Bayesian Nash equilibrium, from John Harsanyi's 1967 to 1968 framework for games of incomplete information. Most of auction theory and the theory of regulation works in this setting.

Weak versus full implementation

The Nobel background document draws a distinction worth knowing. Weak implementation requires that every equilibrium of the game form be optimal according to the target social choice rule. Full implementation additionally requires that every optimal outcome arise as some equilibrium. Maskin's 1999 abstract defines implementation in the full sense: every welfare optimum is a Nash equilibrium outcome and every Nash equilibrium outcome is a welfare optimum.

The revelation principle, explained

The revelation principle is the single most useful result in the subject, and it is easy to state once you have the vocabulary of game forms.

Statement: any equilibrium outcome of any game form can be replicated by an incentive-compatible direct mechanism, that is, a game form in which each player simply reports his or her type and truth-telling is an equilibrium.

The proof idea, as given in the Nobel background document, runs as follows. Fix any game form and any equilibrium of it. In that equilibrium, a player of type t sends some message m(t). Now build a new game form in which each player reports a type, and the new game form computes the message that a player of the reported type would have sent and feeds it into the old rule. If a player could gain by lying about his type in the new game form, he could have gained by sending the corresponding different message in the old one, contradicting the assumption that the old strategies were an equilibrium. So truth-telling is an equilibrium of the new direct mechanism, and it produces the same outcomes.

The practical payoff is that a designer searching for the best possible rules does not have to search over the bewildering space of all conceivable game forms. She can optimize over direct mechanisms, subject to incentive-compatibility constraints, which is a well-defined mathematical problem. Once she finds the best direct mechanism, she can look for a more natural indirect game form that implements the same outcomes. The committee described this as a "seemingly roundabout method" by which problems "that would otherwise have been effectively intractable" were solved.

The principle has a well-known limitation. The new direct mechanism may have other equilibria besides truth-telling, some of them bad. The revelation principle says nothing about those. That gap is exactly what implementation theory, starting with Maskin, addresses.

Worked examples

The following examples show how specific institutions look when written as game forms. Where numbers are not taken from a source, they are labeled hypothetical, and the arithmetic is shown.

Example 1: Plurality voting as a game form

Players: the voters. Actions: each voter writes down one candidate's name. Outcomes: the set of candidates. Rule: the candidate named most often wins, with a fixed tie-breaking rule (say, alphabetical order).

That is a complete game form. It says nothing about who prefers whom. Add a preference profile and you get a game.

The Nobel background document uses plurality rule to illustrate Maskin monotonicity. Maskin monotonicity says that if an alternative a is chosen at one preference profile, and preferences change so that a does not fall in anyone's ranking relative to any other alternative, then a must still be chosen. Consider seven voters and three alternatives, a, b and c:

Initially a is top-ranked by three voters and b and c by two each, so plurality picks a. After the change, voters 6 and 7 now rank b first and a second. Relative to every other alternative, a has not fallen in anyone's ranking: for voters 6 and 7 it actually rose. But b is now top-ranked by four voters (4, 5, 6 and 7) against a's three, so plurality picks b. Since a did not fall yet stopped being chosen, plurality rule fails Maskin monotonicity, and by Maskin's theorem no game form can Nash-implement it. The committee notes that the Borda rule, proposed by Jean-Charles de Borda in 1781, also fails the condition.

Example 2: Approval voting is not strategy-proof

Approval voting is a game form in which each voter marks any subset of candidates as approved, and the candidate with the most approvals wins. The Wikipedia article on Gibbard's theorem gives an example with three voters and candidates a, b and c, ties broken alphabetically. Voter 1 prefers a over b over c. Write a ballot as (approve a, approve b, approve c) with 1 for approve.

• Case 1: the other two voters cast (0,1,1) and (1,1,1). If voter 1 casts (1,0,0), the totals are a = 2, b = 2, c = 2. With alphabetical tie-breaking, a wins. Voter 1 gets her favorite.
• Case 2: the other two cast (0,0,1) and (0,1,1). If voter 1 casts (1,0,0), totals are a = 1, b = 1, c = 2, and c, her least favorite, wins. If instead she casts (1,1,0), totals are a = 1, b = 2, c = 2, a tie between b and c that alphabetical tie-breaking awards to b, her second choice.

So voter 1's best ballot depends on how the others vote. There is no single ballot that is best for her regardless of others. Approval voting has three outcomes and no dictator, and Gibbard's theorem tells us it could not have been strategy-proof. The example shows the theorem's content concretely.

Example 3: The second-price sealed-bid auction

Players: bidders. Actions: each submits a nonnegative number in a sealed envelope. Outcome: the bidder with the highest number receives the item and pays the second-highest number submitted; others pay nothing.

Hypothetical numbers: three bidders value an item at 70, 55 and 40 (in any currency unit). Suppose each bids his or her true value. The bidder valuing it at 70 wins and pays 55, gaining 70 − 55 = 15.

Why is truthful bidding a dominant strategy? Take the bidder with value 70 and suppose the highest competing bid is B.

• If she bids above 70, say 80: when B is below 70, she wins and pays B, exactly as she would have with a bid of 70. When B is between 70 and 80, she wins but pays more than 70, losing money; bidding 70 would have lost the auction at zero cost. Overbidding never helps and sometimes hurts.
• If she bids below 70, say 60: when B is below 60, nothing changes. When B is between 60 and 70, she now loses an item she would have won at a price below her value, giving up a profit of 70 − B. Underbidding never helps and sometimes hurts.

So bidding true value is optimal regardless of what others do. The game form makes honesty a dominant strategy because the price you pay is set by others' bids, not your own; your bid only decides whether you win. Vickrey's 1961 paper also argued that the English (ascending open-outcry) auction and the second-price auction are strategically equivalent, and that the Dutch (descending) and first-price sealed-bid auctions are equivalent to each other.

Example 4: A pivot-style public goods mechanism

The Nobel background document describes the simplest version of the Clarke-Groves mechanism for a yes-or-no decision, such as whether to build a bridge. Each person reports a willingness to pay. The project goes ahead if and only if the total reported willingness to pay exceeds the cost. If it goes ahead, each person pays the difference between the cost and everyone else's total reported willingness to pay.

Hypothetical numbers: a bridge costs 300. Three residents truly value it at 150, 100 and 80. Total value is 330, which exceeds 300, so building is efficient, with a surplus of 330 − 300 = 30.

If everyone reports truthfully, the bridge is built. Payments:

Each resident's net gain equals the total social surplus of 30. That is the trick: each person's payoff is their own value minus cost plus everyone else's reported value, which is the social surplus as computed from reports. Each person therefore wants the decision to maximize true social surplus, and the way to achieve that is to report honestly. A resident's own report affects only whether the bridge is built, not what she pays once it is built.

The table also shows the main drawback. Total payments are 120 + 70 + 50 = 240, which is 60 short of the 300 cost. The Nobel committee notes that this kind of mechanism "does not in general satisfy budget balance," citing Green and Laffont (1979), and that both overfunding and underfunding cause problems: refunding a surplus to participants would destroy truth-telling incentives, while wasting it is inefficient.

Example 5: Bilateral trade and the limits of any game form

The Nobel background document works through a bilateral trade example in detail. A seller A owns an object she values at w; a buyer B values it at v. Both valuations lie between 0 and 1, are private, and are independently and uniformly distributed. Efficiency requires trade whenever v exceeds w.

Consider the double auction game form: both announce a price simultaneously, and if the buyer's bid is at least the seller's ask, they trade at the midpoint. In the linear Bayesian Nash equilibrium the committee describes, the seller asks pA = 2w/3 + 1/4 (if w is no more than 3/4) and the buyer bids pB = 2v/3 + 1/12 (if v is at least 1/4). Trade happens if and only if v exceeds w by at least 1/4.

Computation with hypothetical valuations:

• Seller values at w = 0.3, buyer at v = 0.6. Ask = 2(0.3)/3 + 0.25 = 0.2 + 0.25 = 0.45. Bid = 2(0.6)/3 + 1/12 = 0.4 + 0.0833 = 0.4833. Bid exceeds ask, so they trade at (0.45 + 0.4833)/2 ≈ 0.467. The gap v − w = 0.3 is at least 0.25, as the rule predicts.
• Seller at w = 0.4, buyer at v = 0.6. Ask = 0.2667 + 0.25 = 0.5167. Bid = 0.4833. Bid is below ask, so no trade, even though the buyer values the object more. The gap is only 0.2.

The striking part is what comes next. Myerson and Satterthwaite (1983), and Laffont and Maskin (1979) before them, proved that no incentive-compatible mechanism satisfying voluntary participation can achieve trade exactly when v exceeds w in this setting. By the revelation principle, that means no game form whatsoever can do it. Moreover, the committee notes, this double auction equilibrium actually achieves the best possible expected gains from trade given incentive constraints; it is "incentive efficient." Failures to trade are not a flaw of this particular rulebook. They are a property of the situation.

Example 6: Choosing a reserve price

Myerson's 1981 analysis of optimal auctions asked which game form maximizes a seller's expected revenue. The Nobel background document gives a concrete result: if bidders' values are independently drawn from a uniform distribution between 0 and 100, the optimal reserve price is 50, regardless of how many bidders there are. With that reserve, the four standard auction formats (English, Dutch, first-price and second-price sealed bid) are all optimal for symmetric bidders. The reserve means the object sometimes goes unsold even when a bidder values it above zero and the seller values it at nothing, so the outcome is not efficient in the classical sense, but it maximizes expected revenue within the constraints imposed by private information.

The main theorems in one place

How Gibbard's theorem relates to Gibbard-Satterthwaite

People often blur these two results. The Wikipedia entry on Gibbard's theorem explains that the Gibbard-Satterthwaite theorem is a corollary of Gibbard's more general theorem. Gibbard-Satterthwaite applies to ranked (ordinal) voting rules, where each voter submits a ranking. Gibbard's theorem applies to any game form, including cardinal systems like approval or score voting, where the messages are not rankings. It does not claim that voters must reverse their true rankings to vote strategically, only that a voter's best action depends on what others do.

Monotonicity and the escape through multi-valued rules

The Nobel document points out a sobering consequence of Maskin monotonicity: for single-valued social choice rules over unrestricted preferences, the condition is so demanding that Muller and Satterthwaite (1977) showed it pushes rules toward dictatorship, so that Nash implementation inherits much of the Gibbard-Satterthwaite impossibility. One way out is to allow the target to be multi-valued. The rule that selects all Pareto-efficient alternatives, for example, is Maskin monotonic and can be Nash-implemented. The price is indeterminacy: more than one outcome is acceptable, more than one equilibrium exists, and which one the players reach may depend on bargaining, culture or focal points, a term the committee borrows from Thomas Schelling.

Applications

Game forms are not merely an abstraction. Each of the following areas involves choosing rules without knowing participants' preferences.

Auctions and procurement

Auctions are the most developed application. The committee notes that mechanism design theory "has been used to identify conditions under which commonly observed auction forms maximize the seller's expected revenue," citing Harris and Raviv (1981), Myerson (1981) and Riley and Samuelson (1981). When governments privatize an asset such as radio spectrum, the goal may be to place it with whoever values it most rather than to maximize revenue. Maskin (1992) found conditions under which an English auction does this even when each bidder's value depends on others' private information. The committee also warns that a mistaken initial allocation cannot simply be fixed by later resale, because the Myerson-Satterthwaite result implies that efficient trade between private parties is not guaranteed.

The committee also discusses a cautionary case: Wilson (1979) found that uniform-price auctions for divisible goods have "collusive" equilibria in which bidders split the good at a very low price. According to Paul Klemperer's 2004 book on auctions, cited by the committee, this kind of implicit collusion affected real auctions including the United Kingdom electricity market. The lesson is that a game form must be judged by all of its equilibria, not just the attractive one.

The Wikipedia article on mechanism design notes that advertising auctions run by large online platforms are among the internet-era applications of the theory, alongside routing and electronic commerce.

Elections and voting procedures

Every electoral system is a game form whose message space is the set of possible ballots and whose outcome space is the set of winners. The Gibbard and Gibbard-Satterthwaite theorems tell designers that every reasonable electoral system can, in some circumstances, reward strategic voting. The choice among systems is therefore about which vulnerabilities to accept, not about eliminating manipulation. The committee also notes that voters face coordination problems: a widely held belief that a candidate cannot win can be self-fulfilling, producing multiple equilibria.

Public goods

The bridge example generalizes to shared infrastructure of every kind. The committee lists illustrations: farmers sharing the cost of a common irrigation or drainage system, countries sharing the cost of reducing global warming, and siblings sharing the burden of caring for elderly parents. It also cites a historical argument: that English villages decided earlier than French villages on public goods such as enclosure of open fields and drainage of marshes because French villages required unanimity, which mechanism design shows to be particularly hostile to efficient provision. The committee cites work by Grantham (1980) and Rosenthal (1992) suggesting this may partly explain why English agricultural productivity grew faster between 1600 and 1800.

Regulation

Regulators rarely know a regulated firm's true costs. Baron and Myerson (1982) and Sappington (1982, 1983) modeled regulation as a game form under incomplete information and derived optimal schemes using the revelation principle. In the optimal scheme the regulator trades off extracting rents from the firm against encouraging efficient output, while keeping the firm willing to participate. The committee describes the resulting literature as providing a foundation for comparing price caps with cost-sharing and profit-sharing schemes.

Monopoly pricing and quantity discounts

A seller who can produce many units faces customers whose tastes are private. Mussa and Rosen (1978) and Maskin and Riley (1984) derived the optimal selling mechanism, which involves quantity discounts rather than a single price per unit. Here the game form is a menu of price-quantity pairs, and each customer's choice from the menu reveals information about his or her type.

Engineering and computer science

Game forms appear in engineering wherever autonomous agents share resources. The MIT OpenCourseWare course "Game Theory with Engineering Applications," taught by Asu Ozdaglar and cited in the Wikipedia article, covers mechanism design in that context. Network routing, bandwidth allocation and distributed computing all involve designing rules for agents whose objectives the designer may not control.

Limitations and practical difficulties

Restricted action spaces

Real institutions often cannot use the elaborate message spaces that theory permits. Liad Blumrosen and Michal Feldman's 2013 paper "Mechanism Design with a Restricted Action Space," in Games and Economic Behavior volume 82, studies what happens when bidders can send only a limited number of messages, as when communication is costly. Their work is one of several showing how theoretical optimality degrades when realistic constraints are imposed on the game form.

Pre-existing games

Designers rarely start from scratch. A new mechanism may be layered on top of existing market interactions that participants can still use. Benjamin Roth and Ran Shorrer's 2015 working paper "Mechanism Design in the Presence of a Pre-Existing Game" studies this situation. When players can fall back on an existing game, the designer's freedom to choose the game form is constrained.

Multiple equilibria

As discussed above, a game form with a good equilibrium may also have bad ones. The committee cites Leininger, Linhart and Radner (1989), who found that the double auction has infinitely many (uncountably many) nonlinear equilibria, with welfare ranging from incentive efficient down to zero. Implementation theory addresses this, but the mechanisms that achieve full implementation are sometimes complicated, using integer games or other constructions that seem artificial.

Complexity and plausibility

Maskin-style constructions require each player to report the entire preference profile of everyone, which is plausible only in small groups that know one another well. Theorists have extended implementation to approximate (virtual) implementation, renegotiation-proof equilibria and sequential mechanisms, according to the committee, which lists Matsushima (1988), Abreu and Sen (1991), Maskin and Moore (1999) and Moore and Repullo (1988). Each relaxation trades some robustness for simplicity or some simplicity for robustness.

Budget balance and participation

The VCG examples show that efficiency, truth-telling, budget balance and voluntary participation cannot generally all be had together. The d'Aspremont and Gérard-Varet (1979) mechanism, described by the committee, restores budget balance in a Bayesian setting but violates participation constraints: some individuals, after learning their own type, would prefer not to take part. Mailath and Postlewaite (1990) showed that with voluntary participation and unanimity, the probability of funding a public project can tend to zero as the number of people grows.

Common misconceptions

"A game form is just a game."

No. A game includes preferences; a game form does not. You cannot compute an equilibrium of a game form on its own. The point of the distinction is that the same rules face many possible preference profiles, and the designer must evaluate the rules across all of them.

"If truth-telling is an equilibrium, people will tell the truth."

Not necessarily. The revelation principle guarantees that truth-telling can be an equilibrium of a suitable direct mechanism. Other equilibria may exist. Whether players coordinate on the truthful one is a separate question that implementation theory tries to answer.

"Gibbard-Satterthwaite means voting is meaningless."

The theorem says only that every non-dictatorial rule with at least three alternatives is manipulable in some circumstances. It does not say manipulation is frequent, easy, or profitable in practice, and it does not rank voting systems. Research in computational social choice, such as work reviewed in the Handbook of Computational Social Choice, studies how hard manipulation is to compute for different rules.

"Mechanism design assumes people are perfectly selfish and rational."

The standard models assume players respond to incentives according to a solution concept, typically some form of equilibrium. That is a modeling choice. The theory can incorporate altruism or fairness through preferences, since preferences are whatever the players have. The rules themselves are evaluated against whatever behavior model the analyst adopts.

"The second-price auction must produce less revenue than a first-price auction."

Under the conditions of the revenue equivalence results (including risk-neutral bidders with independent private values), the standard formats produce the same expected revenue. Bidders in a first-price auction shade their bids below their values, which offsets the apparently higher price paid. When those conditions fail, revenue rankings can differ.

"A game form must be a voting rule or an auction."

These are prominent examples, but any procedure that maps participants' actions to outcomes counts: a contract offered by a regulator, a menu of prices, a grading scheme, a matching procedure for students and schools, or a protocol for sharing network bandwidth.

Frequently asked questions

What is the difference between a game form and a mechanism?

In common usage the terms overlap heavily. "Game form" emphasizes the structural object (actions plus outcome rule) without payoffs. "Mechanism" usually signals that the game form was designed with a goal in mind, namely to implement some social choice function. Every mechanism is a game form; the word "mechanism" adds the designer's purpose.

What is an outcome function?

It is another name for the rule g that maps action profiles to outcomes. In mechanism design writing, a mechanism is often described as a pair: a message space and an outcome function.

What does "dictatorial" mean for a game form?

A game form is dictatorial if some single player can force any outcome, whatever the others do. Gibbard's theorem says that strategy-proofness with three or more possible outcomes leaves only dictatorial game forms. A related notion is a duple, a game form with only two possible outcomes; Gibbard's theorem's equivalent statement notes that a non-dictatorial strategy-proof game form must be limited to two outcomes.

Can randomness help?

Gibbard's 1978 paper studied game forms whose outcomes are lotteries. According to the Wikipedia summary of that work, a strategy-proof randomized game form must be a probability mixture of unilateral rules (where one player effectively decides) and duple rules (where only two outcomes are possible). A rule that picks a random voter and implements that voter's favorite, sometimes called random dictatorship, is an example. Randomness widens the possibilities but does not escape the theorem's structure.

Why do auctions get so much attention in this field?

Because they are simple enough to analyze completely, important enough economically to matter, and varied enough in practice (English, Dutch, sealed bid, combinatorial, multi-unit) to test the theory. They also have a natural message space, namely bids, which makes the game form easy to write down.

Is a game form the same as an extensive-form game tree?

A game form can be written in normal form (strategy sets plus outcome function) or in extensive form (a tree of moves with outcomes at the endpoints). Any extensive form can be converted to normal form by treating complete plans of action as strategies. The phrase "game form" is used for both; the essential feature is the absence of payoffs.

Who coined the term?

The idea of a mechanism as a message system with an outcome rule goes back to Hurwicz (1960). The phrase "game form" was used prominently by Gibbard (1973), whose theorem is stated in terms of game forms. Later writers in social choice and mechanism design adopted it widely.

Does a game form need to have finitely many strategies?

Gibbard's original theorem considered finite strategy sets and finite outcomes. Many applications, such as auctions with continuous bids, use infinite strategy sets. The concept itself does not require finiteness; specific theorems may.

Key terms

Summary

A game form strips a game down to its rules: who can do what, and what happens as a result. By removing preferences, it lets designers ask which rules work well across all the preferences people might have. That question produced some of the deepest results in economics and political theory: Gibbard's proof that non-dictatorial game forms with three or more outcomes cannot always reward honesty, the Clarke-Groves discovery that honesty can be made dominant when money can be transferred, the revelation principle that reduces design to the study of truthful direct mechanisms, Maskin's characterization of when all equilibria can be made good, and the Myerson-Satterthwaite proof that some inefficiencies are unavoidable under any rules at all. These results now inform how auctions are run, how regulators write contracts, how public projects are financed, and how electoral systems are compared.

Sources

• Game form (Wikipedia)
• Gibbard's theorem (Wikipedia)
• Mechanism design (Wikipedia)
• Revelation principle (Wikipedia)
• Royal Swedish Academy of Sciences, Scientific background: Mechanism Design Theory (2007)
• Royal Swedish Academy of Sciences, The Prize in Economic Sciences 2007
• Gibbard, Manipulation of Voting Schemes: A General Result, Econometrica (1973)
• Endriss, slides on Gibbard's theorem, University of Amsterdam
• Endriss, Strategic Manipulation lecture slides, University of Amsterdam
• Maskin, Nash Equilibrium and Welfare Optimality, Review of Economic Studies (1999), RePEc record
• Hurwicz, Optimality and Informational Efficiency in Resource Allocation Processes (Cambridge reprint)
• Moulin and Peleg, Cores of effectivity functions and implementation theory (1982), record
• Vickrey, Counterspeculation, Auctions, and Competitive Sealed Tenders (1961), bibliographic record
• Nobel Prize in Economic Sciences 1996, advanced information (Vickrey)