Protocol as Mechanism

Sep 22, 202613 min read

Definition

protocol is a mechanism: a message space for each party and an outcome function that maps every profile of messages to an outcome, where any message outside the legal space maps to the null outcome.

The mechanism induces a game of incomplete information among parties with private types; conformity is a strategy in that game, and the protocol is well-designed when conformity is incentive-compatible (no party gains by deviating) and individually rational (each party prefers admission to the outside option).

Protocols with a native unit of account extend the outcome function with a transfer function that prices moves inside the legal space; protocols without one have only the outcome function and the null outcome as levers, and are mechanisms without money.

Notation

Mechanism:

\Gamma = (M_1, \ldots, M_n, g), \quad g : M_1 \times \cdots \times M_n \to X \cup { \emptyset }

Types and utilities:

\theta_i \in \Theta_i, \quad u_i : X \times \Theta_i \to R, \quad u_i(\emptyset, \theta_i) = 0

Induced payoff with transfer function tᵢ:

U_i(m, \theta_i) = u_i(g(m), \theta_i) - t_i(m)

Fixing protocol (e.g. TCP) as a degenerate mechanism: binary price, no transfers:

t_i(m) = 0 \textrm{ if } m_i \in M_i, \quad t_i(m) = \infty \textrm{ otherwise}

Equivalently, g(m) is the null outcome whenever any mᵢ lies outside Mᵢ.

Incentive compatibility, with conformant strategy mᵢ*:

U_i(m_i^*, m_{-i}, \theta_i) \geq U_i(m_i', m_{-i}, \theta_i) \quad \forall m_i', m_{-i}

Participation (admission) constraint, with outside option ūᵢ:

E[U_i(m^*, \theta_i)] \geq \bar{u}_i

Bridge to the admission-value notation:

V_i = E[u_i(g(m^*), \theta_i)] - c_i

Here cᵢ is the cost of producing legal messages, which standard mechanism theory sets to zero. It is set by the authors and is the one term the protocol fully controls

In mechanism design it is a participation cost: Samuelson, "Competitive Bidding with Entry Costs", Economics Letters 17 (1985); Stegeman, "Participation Costs and Efficient Auctions", Journal of Economic Theory 71 (1996); Celik and Yilankaya, "Optimal Auctions with Simultaneous and Costly Participation", B.E. Journal of Theoretical Economics 9 (2009). Where the cost differs by type, so that conforming is itself informative, it is a signalling cost: Spence, "Job Market Signaling", Quarterly Journal of Economics 87 (1973).

In the economics of standards the same term is the adoption or switching cost: Farrell and Saloner, "Installed Base and Compatibility", American Economic Review 76 (1986); Katz and Shapiro, "Technology Adoption in the Presence of Network Externalities", Journal of Political Economy 94 (1986).

Terms

Mechanism, message space, outcome function: Hurwicz, "Optimality and Informational Efficiency in Resource Allocation Processes" (1960); Hurwicz, "On Informationally Decentralized Systems" (1972); textbook form in Mas-Colell, Whinston and Green, Microeconomic Theory (1995), ch. 23. Size of message spaces: Mount and Reiter, "The Informational Size of Message Spaces", Journal of Economic Theory 8 (1974).
Type / incomplete information: Harsanyi, "Games with Incomplete Information Played by 'Bayesian' Players", Management Science 14 (1967-68).
Incentive compatibility: Hurwicz (1972). Implementation of a social choice rule: Maskin, "Nash Equilibrium and Welfare Optimality" (1977; Review of Economic Studies 66, 1999).
Revelation principle: any mechanism is equivalent to a direct one in which parties truthfully report their types. Gibbard, "Manipulation of Voting Schemes", Econometrica 41 (1973); Myerson, "Incentive Compatibility and the Bargaining Problem", Econometrica 47 (1979). Consequence: every protocol is a truth-telling protocol plus a computation, so "fixing" and "pricing" are not different kinds but different outcome functions.
Participation constraint / individual rationality: Myerson, "Optimal Auction Design", Mathematics of Operations Research 6 (1981). This is the admission condition.
Transfers: Vickrey, "Counterspeculation, Auctions, and Competitive Sealed Tenders", Journal of Finance 16 (1961); Clarke, "Multipart Pricing of Public Goods", Public Choice 11 (1971); Groves, "Incentives in Teams", Econometrica 41 (1973).
Mechanism design without money: Schummer and Vohra, "Mechanism Design without Money", in Nisan et al. (eds), Algorithmic Game Theory (2007), ch. 10; Procaccia and Tennenholtz, "Approximate Mechanism Design without Money", EC (2009). Canonical instance: Gale and Shapley, "College Admissions and the Stability of Marriage", American Mathematical Monthly 69 (1962).
Network protocols as mechanisms: Nisan and Ronen, "Algorithmic Mechanism Design", STOC (1999); Feigenbaum and Shenker, "Distributed Algorithmic Mechanism Design", DIAL-M (2002); Feigenbaum, Papadimitriou, Sami and Shenker, "A BGP-based Mechanism for Lowest-Cost Routing", PODC (2002); Akella, Seshan, Karp, Shenker and Papadimitriou, "Selfish Behavior and Stability of the Internet: A Game-Theoretic Analysis of TCP", SIGCOMM (2002).
Null outcome as enforcement: the outside option in the participation constraint; exclusion is the only sanction a money-free mechanism has.

Fixing and pricing as units

The designer never sets the utilities uᵢ directly; they are the players' private types.

The designer controls only the outcome function g and the transfer function t, and the induced payoff table follows from both:

U_i(m) = u_i(g(m), \theta_i) - t_i(m)

So a protocol writes the payoff table in one of two ways, fixing or pricing.

Fixing writes it indirectly, through the message spaces and g alone: entries depend on exogenous, unobserved uᵢ, so the protocol shapes the table (exclusion zeroes a row; the strategy space decides which columns exist) without being able to read its numbers.

Pricing writes entries directly, through t, in a common unit the designer chose and can read. Fixing writes the table in an unknown unit; pricing writes it in a known one.

Where the payoff and cost are non-monetary

Any mechanism without transfers that is strategy-proof over an unrestricted domain of preferences with three or more outcomes is dictatorial: Gibbard (1973) and Satterthwaite, "Strategy-proofness and Arrow's Conditions", Journal of Economic Theory 10 (1975).

This means that where the payoff and cost are non-monetary and the protocol has no other unit it controls, a protocol can be incentive-compatible only if the preference domain is restricted, which is exactly the coordination-game case of aligned preferences over interoperation (cf. Moulin, "On Strategy-Proofness and Single Peakedness", Public Choice 35 (1980), for the standard restricted-domain result).

Where the payoff and cost are monetary, the protocol can write entries whose sign it knows regardless of type, and incentive compatibility becomes available on cooperation domains. The mechanism-first view therefore predicts the genus/species split rather than dissolving it: a unit the protocol controls is what lets it leave the coordination domain.

Reputational, social and political capital as the unit

These theorems do not require money. They require a unit that is comparable across players and that the mechanism can move. Reputational, social and political capital can serve, and the theory says exactly how far each gets you, on a scale set by what the mechanism can do with the unit.

The strongest case is a unit that can be moved between players. Then the full apparatus applies: Vickrey-Clarke-Groves payments, budget balance, quasi-linear welfare. Social capital in Coleman's formulation is explicitly of this kind: obligations held as "credit slips" that can be called in and passed on. Coleman, "Social Capital in the Creation of Human Capital", American Journal of Sociology 94 (1988). Bourdieu's account of capital as convertible between economic, social and cultural forms at rates set by the field is the same claim from the other direction: a convertible unit is a transferable one. Bourdieu, "The Forms of Capital", in Richardson (ed.), Handbook of Theory and Research for the Sociology of Education (1986). Political capital as favours owed and votes deliverable is transferable in this sense, though the conversion rates are illiquid.

Reputation in the usual sense is weaker than this: it cannot be paid to another player, only lost. There are two ways to model a unit that can be destroyed but not transferred, and the second is stronger.

The first treats the loss as a payment that is destroyed rather than transferred. This is the "money burning" case, characterised by Hartline and Roughgarden, "Optimal Mechanism Design and Money Burning", STOC (2008): incentive compatibility is achievable, but the burn happens on the equilibrium path and the welfare cost equals what is burned. That result assumes quasi-linear utility in the burned unit, which reputation need not satisfy, and it fits protocols where the cost is paid whether or not anyone deviates: proof-of-work is on-path money burning in exactly this sense.

The second obviates both the quasi-linearity assumption and the welfare loss by not treating reputation as a unit at all. In a repeated game the thing a deviator loses is the continuation value of admission, which the protocol controls by exclusion, and which is already denominated in the player's own utility. With per-period admission value Vᵢ, discount factor δ, and one-shot gain from deviation gᵢ, deviation is deterred when:

g_i \leq \frac{\delta}{1 - \delta} V_i

Folk theorem for the general result: Fudenberg and Maskin, "The Folk Theorem in Repeated Games with Discounting or with Incomplete Information", Econometrica 54 (1986). Optimal punishment structure: Abreu, "On the Theory of Infinitely Repeated Games with Discounting", Econometrica 56 (1988). The version for strangers, where each player meets a different counterparty each period and the record must travel: Kandori (1992), and Ellison, "Cooperation in the Prisoner's Dilemma with Anonymous Random Matching", Review of Economic Studies 61 (1994). Survey: Mailath and Samuelson, Repeated Games and Reputations (2006).

On this model the punishment is off the equilibrium path: nothing is burned when everyone conforms, so there is no welfare cost in equilibrium, and no assumption is needed about how reputation enters utility because reputation is not an argument of utility but a stock of expected future admission. Slashable stake with a public ledger is this structure with the continuation value made explicit and posted up front. So reputational protocols can leave the coordination domain without paying in destroyed value, provided three things hold: interaction is repeated, players are patient enough for the inequality to bind, and deviation is recorded and propagated. The third condition is the ledger requirement below.

None of this works unless the mechanism can read the unit, which means deviation must be recorded and the record must reach future counterparties. This is the community-enforcement condition: Kandori, "Social Norms and Community Enforcement", Review of Economic Studies 59 (1992); Milgrom, North and Weingast, "The Role of Institutions in the Revival of Trade: The Law Merchant", Economics and Politics 2 (1990); Greif, "Contract Enforceability and Economic Institutions in Early Trade: The Maghribi Traders' Coalition", American Economic Review 83 (1993). Formal reputation mechanisms for online markets: Dellarocas, "The Digitization of Word of Mouth", Management Science 49 (2003); Resnick and Zeckhauser, "Trust Among Strangers in Internet Transactions", in Baye (ed.), The Economics of the Internet and E-Commerce (2002). The ledger is the accountability layer; without it the unit is not controllable and the protocol drops back to the fixing-only case.

At the far end, esteem that is held privately, not recorded, and not comparable across players is not a unit the mechanism controls at all. Protocols that rely on it are in the Gibbard-Satterthwaite regime and must confine themselves to restricted domains.

Ordering the cases, from most to least mechanism power:

Transferable money or tokens.
Transferable social obligations.
Repeated interaction with a public ledger, whether the stake is reputation or slashable deposit.
Unrecorded esteem.

Where the payoff and cost are denominated in the first two, the protocol can reward and punish; in the third it can threaten exclusion, which suffices without on-path cost when players are patient; in the fourth it can only exclude with no memory, so the threat has no bite.

Patience decomposed: time preference, continuation, and punishment incidence

The deterrence inequality applies to whoever acts. Where the actor is an agent whose horizon and stake differ from the principal's, "patience" is three separate terms, and only one is time preference. With gᵢ the one-shot gain, Vᵢ the per-period admission value, αᵢ the share of continuation value the acting agent bears, βᵢ pure time preference, and pᵢ the probability the game continues for this actor.

Notation

g_i \leq \frac{\delta_i}{1 - \delta_i} \alpha_i V_i, \quad \delta_i = \beta_i p_i

Terms

Pure time preference, βᵢ, is the standard discounted-utility term: Samuelson, "A Note on Measurement of Utility", Review of Economic Studies 4 (1937); Frederick, Loewenstein and O'Donoghue, "Time Discounting and Time Preference: A Critical Review", Journal of Economic Literature 40 (2002). The sharper concept for protocol is time inconsistency: a discretionary actor's future self cannot be trusted to honour the present self's plan, so a fixed rule outperforms discretion even for a single actor. Strotz, "Myopia and Inconsistency in Dynamic Utility Maximization", Review of Economic Studies 23 (1955); Kydland and Prescott, "Rules Rather than Discretion: The Inconsistency of Optimal Plans", Journal of Political Economy 85 (1977). This is the fixing-versus-discretion argument restated as a property of one player over time.

The continuation probability, pᵢ, is Axelrod's "shadow of the future", The Evolution of Cooperation (1984); folding continuation probability into the discount factor is standard, Fudenberg and Tirole, Game Theory (1991), ch. 5. Term limits and exit options push pᵢ toward zero and the model predicts end-game defection: Besley and Case, "Does Electoral Accountability Affect Economic Policy Choices? Evidence from Gubernatorial Term Limits", Quarterly Journal of Economics 110 (1995). Finite-horizon unravelling: Selten, "The Chain Store Paradox", Theory and Decision 9 (1978); the reputation-based repair is Kreps, Milgrom, Roberts and Wilson, "Rational Cooperation in the Finitely Repeated Prisoners' Dilemma", Journal of Economic Theory 27 (1982).

The internalised share, αᵢ, is a question of punishment incidence: whether the loss from deviation lands on the actor or on someone else. It is the ownership-share parameter of Jensen and Meckling, "Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Structure", Journal of Financial Economics 3 (1976). The cleanest formal analogue is limited liability: the agent's downside is capped at what they personally hold (the office, the seat) while the principal's loss is not. Sappington, "Limited Liability Contracts between Principal and Agent", Journal of Economic Theory 29 (1983); Innes, "Limited Liability and Incentive Contracting with Ex-ante Action Choices", Journal of Economic Theory 52 (1990). Structurally, the reason αᵢ is low is that the party's or state's standing is a common-pool resource each agent can draw down for private gain, which makes this a commons problem and makes monitoring and graduated sanctions the repair: Ostrom, Governing the Commons (1990). Moral hazard in Holmström's sense is not the right label, since it bundles incidence with unobservability, and unobservability is the condition the ledger requirement has already removed. Political agency versions of the incidence problem: Barro, "The Control of Politicians: An Economic Model", Public Choice 14 (1973); Ferejohn, "Incumbent Performance and Electoral Control", Public Choice 50 (1986).

Political protocol under this definition

Diplomatic protocol proper (precedence, forms of address, sequencing of recognition) is fixing on a coordination domain among players who are the ideal case for the repeated-game model: states are long-lived, few, patient, and keep an exceptionally good ledger. Deviation is legible by design, since a snub is only a snub because the expected move was fixed, and the punishment is reciprocal exclusion. All three terms of the inequality are high because the state is the actor. Political capital as a transferable unit is also grounded: logrolling is vote-trading in a common unit, Buchanan and Tullock, The Calculus of Consent (1962), ch. 10, and Coleman's credit slips (1988) generalise it.

Norms of political forbearance, by contrast, are enforced at the level of the individual politician, for whom pᵢ is the electoral cycle and αᵢ is small because the loss from deviation falls on the party, the office or the state. Levitsky and Ziblatt, How Democracies Die (2018), describe the failure of mutual toleration and institutional restraint; on this model those norms fail not because anyone's time preference changed but because punishment incidence falls on the party, the office or the state rather than on the politician who deviates (αᵢ is low, and the politician's liability is limited to their own seat), the electoral cycle and exit into private life truncate the horizon (pᵢ is low), and the ledger (press, opposition memory, institutional record) has stopped propagating deviation, so the common-pool standing of the institution is drawn down by each agent in turn. Diplomatic protocol is robust and forbearance norms are fragile for the same reason, read from opposite sides of one inequality.

(Header image generated with nano banana.)

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