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Econ-193 · economics · Economics

Voluntary participation over time

LLM status Open unverified

Statement

Original subject group: Mechanism design and market design

Audit classification: Published research agenda. The survey’s §8 discusses participation and initial bonding. Wealth-limited escrow design is an editorial extension; finite-history optimization alone is not a new existence problem.

Definitions and precise research target

Model and research target. Let one seller repeatedly allocate a service to $n$ agents over $T$ periods. Agent $i$'s type follows a known finite Markov chain and yields quasilinear discounted utility, with common discount factor $\beta\in(0,1]$. A direct mechanism chooses feasible allocation $x_t$ and nonnegative payment $p_{it}$ from report histories. Agent $i$ has observable initial liquid wealth $B_i$, cannot borrow, and may exit after any private history with no future allocations or payments. Impose the pathwise constraint $\sum_{s\leq t}p_{is}\leq B_i$, dynamic incentive compatibility against every reporting-and-exit strategy, and sequential participation \[ E\left[\sum_{s=t}^{T}\beta^{s-t} (v_i(x_s,\theta_{is})-p_{is})\mid h_{it}\right]\geq0 \] at every truthful private history $h_{it}$. Characterize revenue-optimal mechanisms and the loss relative to mechanisms requiring only initial participation, as functions of wealth, type persistence, and horizon. Determine whether limited refundable escrow or capped deposits recover part of that loss, stating the cash and exit rules exactly before comparing mechanisms. An existence or characterization result must include off-path incentive restrictions and histories reached after other agents' reports.

Definitions and maintained assumptions (editorial unless a source definition is named). The known joint initial and Markov type law, allocation feasibility and valuation functions are primitives; agents know their own types and the specified public information. Exit at the beginning of period $t$ eliminates future trades and payments and gives normalized future utility zero; previously received service and paid fees remain sunk. With nonnegative payments and no wealth replenishment, $\sum_{s\le t}p_{is}\le B_i$ is a pathwise liquidity constraint in nominal currency, distinct from discounted utility. Dynamic IC compares truth-and-continuation with every nonanticipating reporting and exit strategy at every feasible private history, using a specified off-path posterior rule or ex-post robustness. For finite types/horizon, randomized mechanisms with bounded payments can be encoded by finitely many history variables and linear IC/IR/liquidity constraints; mere numerical computability is not the open issue. Seek structural formats, bounds uniform in horizon and polynomial representation/computation under explicitly encoded primitives. Refundable escrow needs a cash-balance process recording deposits, interest, forfeiture and refunds; it cannot simply be added to a no-refund cumulative-payment constraint.

Variants and assumption changes

  1. No escrow (editorial). Study a single buyer with two Markov types, nonnegative fees, zero future exit payoff and initial wealth $B$. Characterize horizon-uniform structural policies and the revenue gap from initial-only participation.
  2. Refundable deposit (editorial). Permit an initial deposit at most $B$, immediate refunds on exit and specified deductions. Compare revenue under the same cash-flow feasibility and truthful-exit constraints; unrestricted bonding is a different model.

References and source audit

  1. §8 printed p.50 in June2018 revision. Supports the stated research area; exact displayed specification is editorial. DOI publication is2019; inspected revision is June2018. https://cowles.yale.edu/sites/default/files/2022-09/d2102-r.pdf
  1. Dirk Bergemann; Juuso Välimäki. 2019. Dynamic Mechanism Design: An Introduction. Published JEL57(2),235–274 (2019); inspected June19,2018 Cowles revision §8, printed pp.49–51. https://cowles.yale.edu/sites/default/files/2022-09/d2102-r.pdf. DOI: https://doi.org/10.1257/jel.20180892.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-193-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-193

Voluntary participation over time A horizon-uniform solution for persistent types and refundable escrow

Problem ID: D82-189. Status: Unresolved; restricted results and reductions.

The research target and the absorbing-type restriction

The full target concerns dynamic incentives, nonnegative payments, liquidity, and exit after every private history. Bergemann and Välimäki [bv, Section 8, printed p. 50] discuss the distinction between initial and later participation, including initial bonds. The wealth and refund rules below are additional model assumptions.

Consider one buyer, $T\geq1$ periods, discount factor one, and a private type $\theta\in\{\ell,h\}$ with $0<\ell<h$, drawn once and then constant. This is a two-state Markov chain with both states absorbing. The probability of $h$ is $q\in(0,1)$. In each period the seller can provide service amount $x_t\in[0,1]$ at zero cost; the buyer's value is $\theta x_t$. Payments are nonnegative, with pathwise total at most observable wealth $B\geq0$. Revenue means total payments received. The seller commits to the mechanism. Exit before any period gives zero future service and payment, with past trades sunk. Later reports are allowed but may be ignored by a mechanism. This divisible-service specialization gives deterministic implementations of fractional quantities; lotteries over indivisible services are another version.

We solve the revenue problem exactly in this restriction for every $T,B,\ell,h,q$, prove that sequential participation causes no revenue loss relative to initial-only participation, and prove an equivalence for genuinely refundable escrow. Neither conclusion is asserted for changing Markov types.

Reduction to a two-option screening problem

For any dynamically incentive-compatible mechanism, let $X_\theta$ be the expected total service and $P_\theta$ the expected total payment under truthful continuation from initial type $\theta$. Then $0\leq X_\theta\leq T$, $0\leq P_\theta\leq B$. A buyer can imitate the entire reporting strategy of the other absorbing type, including its responses to public randomization. Thus initial dynamic IC and IR imply \begin{align} \ell X_\ell-P_\ell&\geq0,\qquad hX_h-P_h\geq0,\label{wsref-0-69-72}\\ hX_h-P_h&\geq hX_\ell-P_\ell,\label{wsref-0-68-69-63}\\ \ell X_\ell-P_\ell&\geq\ell X_h-P_h.\label{wsref-0-6c-69-63} \end{align} These are necessary even if the mechanism uses all $T$ report histories and randomizes. Using them as an upper bound does not presuppose that every static menu is sequentially implementable.
Lemma (An upper envelope for revenue): If $B\geq\ell T$, every dynamically IC mechanism satisfying initial participation has revenue at most \[ \max_{0\leq x\leq T} \left((1-q)\ell x+q\min\{B,hT-(h-\ell)x\}\right). \] For every $B$, revenue is at most $B$.
Proof: Initial low-type IR gives $P_\ell\leq\ell X_\ell$. High-type IC then gives $P_h\leq hX_h-(h-\ell)X_\ell\leq hT-(h-\ell)X_\ell$. Combine with $P_h\leq B$ and substitute $x=X_\ell$. The expression is an upper bound regardless of any omitted constraints, including low-type IC. The pathwise budget bound implies $P_\theta\leq B$ for both types and hence expected revenue at most $B$. ∎

Exact revenue and an implementation after every history

For $B\geq\ell T$, define \[ b_H=\min\{B,hT\},\qquad x_0=\max\left\{0,\frac{hT-B}{h-\ell}\right\}\in[0,T]. \]
Theorem (Horizon-uniform optimal menu): The optimum revenue with dynamic IC, pathwise liquidity, and sequential participation is \[ R^*(T,B)= \begin{cases} B,&0\leq B\leq\ell T,\\[2mm] \max\{\ell T,\ q b_H+(1-q)\ell x_0\},&B\geq\ell T. \end{cases} \] The same value is optimal when only initial participation is required and future exit is disallowed. An optimal mechanism has at most two initial options, collects all fees immediately, commits the service schedule, and ignores subsequent reports. For rational $B,\ell,h,q$ and a positive integer $T$ encoded in binary, its description has size polynomial in their encodings.
Proof: If $B\leq\ell T$, sell $T$ units of service for an initial fee $B$ to both types. The buyer can pay, both types' initial utility is nonnegative, and later all remaining service is free. This attains the universal bound $B$.

Now let $B\geq\ell T$. The objective in Lemma Lemma (An upper envelope for revenue) is piecewise linear. If $B<hT$, its slope is $(1-q)\ell>0$ on $[0,x_0]$ and $\ell-qh$ on $[x_0,T]$. If $B\geq hT$, it has the single slope $\ell-qh$ and $x_0=0$. Its maximum is therefore attained at either $x_0$ or $T$. At $T$ its value is $\ell T$; at $x_0$ it is $q b_H+(1-q)\ell x_0$. This proves the proposed upper bound even for initial-only participation.

To attain the value $\ell T$, offer everyone all $T$ services for initial price $\ell T$. To attain the other candidate, offer total quantity $x_0$ at initial fee $\ell x_0$ and total quantity $T$ at initial fee $b_H$. The low type gets zero from the first option and $\ell T-b_H\leq0$ from the second. The high type gets $(h-\ell)x_0$ from the first and $hT-b_H$ from the second; these are equal by the definition of $x_0,b_H$. Select the second option when the high type is indifferent. Both fees are nonnegative and at most $B$. Implement quantity $x$ by providing one unit in periods $1,\ldots,\lfloor x\rfloor$, the fractional remainder in the next period when needed, and zero thereafter.

At the initial private history the two IC inequalities just checked and nonnegative utility make truth and participation optimal. After any initial report, including a false one, the fee is sunk, the schedule is fixed, and no further payments are due. Continuing yields nonnegative future utility for either true type; exiting cannot improve it. Subsequent reports do not change the schedule, so no reporting or reporting-and-exit strategy after any feasible history improves utility. This verifies off-path continuation incentives directly. A deviation at the initial history is only a choice among the two options and exit, which was already checked. Thus the upper bound is attained by a sequentially admissible mechanism, proving equality of the two participation benchmarks. Encoding one quantity and one upfront fee per option plus the simple service rule has the claimed size. ∎

Weak IC permits the stated tie selection, just as a direct truthful mechanism may select a recommended action among utility-equivalent ones. No strict implementation claim is being made.

Corollary (Liquidity regions and the role of persistence): For $B\geq hT$, the optimum is $T\max\{\ell,qh\}$. For $B\leq\ell T$, it is $B$. For $\ell T<B<hT$, either the pooled low-price contract is optimal, or the optimal low-type quantity is $(hT-B)/(h-\ell)$ and the high type pays the entire budget $B$. In this absorbing-type family the sequential-participation revenue loss is identically zero for every horizon and every wealth level.
Proof: Substitute the respective ranges into Theorem Theorem (Horizon-uniform optimal menu). The zero-loss statement is its equality of the two feasible optimum values. ∎

This identifies a boundary case where a positive dynamic participation loss cannot be universal. It does not say that the loss vanishes when new private information arrives: the imitation argument then involves conditional type laws, and freezing an initial menu generally loses important allocation possibilities.

What completely refundable escrow can and cannot add

Now allow any finite type process and any finite horizon, still one buyer, discount factor one, no other uses of cash, no borrowing, and initial wealth $B$. Define an escrow mechanism as follows. At the initial date the buyer may place $B$ into an account owned by the buyer. This is not seller revenue. At each participating period the mechanism may deduct a nonnegative fee $p_t$ from the account, at most its current balance; the deduction is seller revenue. There is no interest, forfeiture, top-up, or additional deposit charge. Undeducted funds are refunded immediately upon exit, and at the end if the buyer stays. The balance is $E_t=B-\sum_{s\leq t}p_s$. The buyer can reject the initial contract. All policies and deductions may depend on reports and public randomization.
Theorem (Refundable-escrow equivalence): Under these exact cash and exit rules, adding the escrow account does not expand the achievable allocation, net-payment, or revenue laws of sequentially participating, dynamically IC mechanisms with pathwise nonnegative fees of total at most $B$. This holds for arbitrary finite type dynamics, including after every feasible off-path history.
Proof: Given an escrow mechanism, simulate its report and randomization history without a deposit. On every participating path charge the same nonnegative deduction $p_t$ as an ordinary contemporaneous fee. Before that charge the buyer has liquid wealth $B-\sum_{s<t}p_s$, exactly the simulated account balance, so the fee is feasible. If she exits, ordinary future fees stop; in escrow her account is refunded. For any reporting-and-exit strategy, her total net payment in escrow is exactly the deductions made before exit: the initial $B$ outflow and the final remaining-balance refund cancel. With discount factor one, utility and seller revenue therefore agree path by path in the two implementations. The same comparison applies conditional on any feasible history, subtracting the refundable balance as owned wealth. All IC and sequential participation inequalities are preserved.

Conversely, given a no-escrow mechanism, deposit $B$ initially and make its original fees the deductions. The cumulative-fee constraint ensures that every deduction is feasible. Refund the remaining balance on exit or termination. The same cancellation proves equality of utility and revenue for every strategy and every history. Therefore both directions preserve the attainable laws and their incentive restrictions. ∎

The account is a liability, not extra seller revenue. Exit forfeiture, interest, discounting of deposit opportunity costs, outside spending, or wealth replenishment would invalidate some steps of this path-by-path equivalence and must be modeled separately. In particular this theorem does not equate fully refundable escrow with the forfeitable bonds discussed in the source.

Remaining task

The closed-form persistent-type solution and the escrow equivalence go beyond finite-history linear-program existence, but leave the principal challenge of changing privately observed Markov types, multiple buyers, nontrivial off-path beliefs, and discounted cash flows unresolved. Horizon-uniform approximation or structural results for those cases would be needed to address the complete catalogue target.

References

  • [bv] D. Bergemann and J. Välimäki (2019). Dynamic Mechanism Design: An Introduction. Journal of Economic Literature 57(2), 235--274. Consulted version: Cowles Discussion Paper 2102R, revision June 19, 2018, Section 8, printed pp. 49--51, particularly p. 50. https://cowles.yale.edu/sites/default/files/2022-09/d2102-r.pdf.

Completion Estimate

40%

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