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Asymmetric Information in Dynamic Lending Games

Kristijan Atanasov, Frederik Mallmann-Trenn, Carmine Ventre

arXiv:2610.13759v1cs.GT

TL;DR

The paper asks how access to aggregate rival-supply information affects lender payoffs when supply is reusable and demand varies. It compares information assignments in a dynamic lending game and finds that the payoff ranking can depend on both demand and which lender receives information, even when aggregate lending is unchanged.

  • Problem

    With reusable supply, a lender’s fee affects both its current payoff and its ability to serve future borrowers, motivating study of how rival-supply information and its allocation affect payoffs as demand changes.

    Source: §1 · p. 2

  • Method

    The paper analyzes observation-constrained pure stationary equilibria in a stochastic lending game, comparing regimes in which none, some, or all lenders observe aggregate rival available supply.

    Source: §1 · p. 2

  • Results

    For two lenders, informing only the larger can yield higher total payoff at low demand, while informing only the smaller can rank highest near supply; payoff gains can occur without increased aggregate lending.

    Source: §1 · pp. 2–3, §4 · p. 13

  • Takeaways & Limitations

    Information reassignment can change lender payoffs through fees and loan allocation even when total lending stays unchanged.

    Source: §1 · p. 2, §3 · p. 7

    Loan durations are assumed exponential, a simplification for rentals with scheduled end dates.

    Source: §2.2 · p. 4

Abstract

from arXiv · show

We study a dynamic lending game with limited but reusable supply. Borrower arrivals and loan durations are stochastic. Demand is allocated cheapest-first, subject to available supply, so a lender's quote affects both its current payoff and its ability to serve future borrowers. Motivated by industry practice, we study this problem through the lens of information disclosure: Lenders observe either only their own available supply or also their rivals' aggregate available supply. We ask how providing this information to all, some or none of the lenders affects payoffs as demand varies relative to supply, and how these effects change when a fixed total supply is divided among different numbers of lenders. Interestingly, in the case of asymmetric information disclosure, total payoffs can increase, while total lending remains the same. We identify sufficient mathematical regimes in which this phenomenon occurs. Who receives supply information matters differently as demand rises. For the case of two lenders, informing only the larger lender yields higher total lender payoff at lower demand, while informing only the smaller lender can yield the highest total lender payoff at higher demand. For a platform deciding which lenders can observe rival supply, the results suggest that disclosure can change the fees paid by borrowers without changing how much is lent. Wider access to supply information need not lower those fees: its effect depends on who receives the information and the balance between demand and supply.

1 Introduction

The paper studies how access to aggregate rival-supply information shapes competition among lenders with reusable supply. It finds that the payoff effects depend on who receives information and on demand relative to supply.

1 Introduction

  • Reusable supply creates a trade-off: winning a borrower produces an immediate fee but temporarily reduces a lender’s ability to serve later demand.The paper motivates this setting with securities lending and rentals of accommodation, equipment, and computing capacity.Source: §1 · p. 2
  • The game models stochastic borrower arrivals and loan returns, with the lowest available fee winning and the selected lender supplying the unit.Lenders may observe only their own available supply or also aggregate rival supply; no borrower is served when supply is unavailable.Source: §1 · p. 2
  • The analysis compares information regimes as borrower demand, the division of fixed supply, and the permitted fee set vary.The authors identify sufficient parameter conditions for some payoff rankings and examine how supply-information allocation interacts with unequal supply and demand pressure.Source: §1 · p. 2
  • Informing only the larger lender can yield higher total payoff at low demand, while informing only the smaller can rank highest when demand is close to supply.For two lenders, moving information access to the smaller lender can strictly benefit both, including the larger lender that loses access; these gains do not require more aggregate lending.Source: §1 · pp. 2–3
  • Unlike work where shared occupancy information can intensify competition, this paper examines how information assignment interacts with unequal supply and demand pressure.The model serves each arrival whenever supply is available and uses a common, fee-independent loan-return rate, so lending is unchanged by pricing, information, and supply division.Source: §1 · p. 3
  • The analysis checks unilateral deviations in observation-constrained pure stationary strategies without specifying a learning algorithm.The authors distinguish pricing incentives from exploration or updating rules and caution that higher fees alone do not establish learned collusion.Source: §1 · p. 4

2 Model

The model is a continuous-time lending game in which lenders quote fees using their permitted supply observations. Reusable units move between available and on-loan states, and payoffs account for future bookings and returns.

2 Model

  • Borrowers arrive at rate λ and request one unit; loans have independent exponential durations with rate μ, and lenders maximize discounted payoff using pure stationary strategies.The strategies assign fees based on observed available supply, with lenders observing their own supply and potentially aggregate rival supply.Source: §2 · p. 4

2.1 Lenders and supply

  • Each lender owns a fixed number of indivisible units; available supply is the portion currently lendable, while the remainder is on loan and returns to its owner.The starting state has all units available, and lenders need not observe the full supply state.Source: §2.1 · p. 4

2.2 Lending mechanism

  • Lenders quote from a common finite fee set using their current supply observations, and the lowest available quote receives the borrower.Ties are allocated uniformly; booking removes one available unit and a completed loan returns it to the same lender. Exponential durations simplify scheduled-end rentals because return rates depend on outstanding loans, not time already elapsed.Source: §2.2 · pp. 4–5

2.3 State transitions

  • For fixed strategies, bookings and returns generate a continuous-time Markov chain over supply states.A booking lowers the selected lender’s available supply by one, while returns raise that lender’s available supply; arrivals with no available supply do not change the state.Source: §2.3 · p. 5

2.4 Payoffs and state values

  • A lender’s value is the expected discounted payoff from fees received at bookings, calculated from the state-transition process and its reward rates.The payoff can also be expressed as discounted time spent in each state weighted by the lender’s reward there.Source: §2.4 · pp. 5–6

2.5 Equilibrium

  • Equilibrium requires that no lender can improve its payoff by changing quotes at any or all permitted observations while rivals’ strategies remain fixed.Deviations are evaluated from the fully available starting state and include their effects on subsequent bookings and returns; when multiple pure stationary equilibria exist, the model selects one with the highest total discounted payoff.Source: §2.5 · p. 6
  • An informed lender observing zero aggregate rival available supply must quote the maximum permitted fee in equilibrium.At that observation, raising a lower fee does not change allocation probability or transition rates and strictly increases payoff; this need not hold for an uninformed lender.Source: §2.5 · p. 6

3 Results

The results examine how information disclosure and supply division shape lender payoffs and total lending as demand varies, including how fragmenting ownership can affect total payoff as pricing shifts between lenders.

3 Results

  • Expected discounted successful bookings are invariant across admissible strategies, information regimes, and divisions of fixed total supply.The aggregate available-supply process has the same distribution under each arrangement, so higher payoffs cannot be explained by increased aggregate lending.Source: §3 · p. 7

3.1 Information assignment

  • With two lenders, informing the larger lender yields higher payoffs at lower arrival rates, while informing only the smaller can yield the highest payoff at higher arrival rates.For supplies (2, 1), a sufficient parameter region also gives OA higher total payoff than AO.Source: §3.1 · pp. 7, 11

3.1.1 Numerical comparison

  • For supplies (2, 1), fees {0.75, 1}, and discount rate 0.1, equilibria were checked over arrival and return rates from 0.1 to 5.0 in increments of 0.1.At each parameter pair, the analysis checked every admissible unilateral deviation, accepting payoff gains of at most 10^-10.Source: §3.1.1 · p. 11

3.1.2 Pareto improvements

  • In an unequal-supply lending game, informing only the smaller lender rather than only the larger strictly improves both lenders’ payoffs: AO gives 11.7404 and 4.0717; OA gives 11.9532 and 5.4552.Source: §3.1.2 · p. 12

3.2 Ownership fragmentation

  • Splitting the smaller lender’s two units into two independent informed lenders can raise total equilibrium payoff without increasing lending.The comparison is between (2, 2) under OA and (2, 1, 1) under OAA; the proposition establishes a sufficient parameter region.Source: §3 · p. 7, §3.2 · p. 12
  • The verified equilibria quote pL when competing, but pH when alone; lender 1 also quotes pH with one available unit after the split.In the (2, 2) game, lender 1 quotes pL whenever available; in the (2, 1, 1) game, it quotes pH if x1=1 and pL if x1=2.Source: §3.2 · p. 12
  • When λ≥4(μ+δ), splitting lender 2 raises total payoff by shifting high-price bookings to states with exactly one unit available overall.The proof uses symmetric supply transitions to establish z1,0>z0,2; the split removes lender 2’s high fee when it alone has two units, while lender 1 quotes pH with one unit available.Source: §3.2 · pp. 12–13

4 Conclusion and Future Work

Within observation-constrained pure stationary equilibria, the payoff ranking between informing only the larger or smaller lender depends on demand relative to supply. Reassigning information can change payoffs without changing total lending.

4 Conclusion and Future Work

  • At low demand, informing only the larger lender can yield higher total payoff than informing only the smaller, but this ranking can reverse near supply.With other parameters unchanged, informing only the smaller lender then yields the highest total payoff among all four information regimes.Source: §4 · p. 13
  • With two permitted fees, sufficient conditions give higher total payoff under OA than AO for any larger-lender supply when the smaller lender owns one unit.Reassigning information can also benefit both lenders, and splitting an informed lender can raise total payoff.Source: §4 · p. 13
  • At fixed total supply, arrival and return rates, and discount rate, payoff differences come from fees and discounted state times, not additional lending.Source: §4 · p. 13
  • Future work could allow loan end dates to be known at booking, consider a continuous fee interval, and extend comparisons to more lenders.Continuous fees would permit arbitrarily small undercuts and require fresh analysis of equilibrium existence and payoff rankings.Source: §4 · p. 13

A Information Assignment: Proofs and Equilibrium Verification

The proofs verify equilibria by constructing the transition and reward objects for each strategy profile, calculating payoffs, and checking admissible unilateral deviations.

A Information Assignment: Proofs and Equilibrium Verification

  • For each strategy profile, the proof constructs Q(σ) and r_i(σ), computes V_i(σ) from (1), and compares payoffs across admissible unilateral deviations.Source: A Information Assignment: Proofs and… · p. 16
  • The first table block reports one lender’s best responses to each rival strategy, and the second checks the rival’s responses to the distinct best responses found.Source: A Information Assignment: Proofs and… · p. 16
  • Payoff comparisons use exact arithmetic, while displayed values are rounded to four decimals and bold entries mark equilibrium strategies and payoffs.The next-best payoff is the highest payoff from any different remaining strategy against the same rival strategy.Source: A Information Assignment: Proofs and… · p. 16

A.1.1 Equilibrium verification for 𝜆= 1

For supplies (2, 1) and fees {0.75, 0.875, 1} at λ=μ=1, the appendix verifies equilibria under AO and OA and notes equilibria under OO and AA.

A.1.1 Equilibrium verification for 𝜆= 1

  • Under AO, the unique equilibrium has lender 1 quote 1 when alone and 0.75 when competing, while lender 2 always quotes 0.75.Source: §A.1.1 · p. 16
  • Under OA, the unique equilibrium has lender 1 always quote 0.75, while lender 2 quotes 1 when alone and 0.75 when competing.Source: §A.1.1 · p. 17
  • Both lenders always quoting 0.75 is an OO equilibrium, while under AA they quote 1 when alone and 0.75 when competing.Source: §A.1.1 · p. 17

A.1.2 Equilibrium verification for 𝜆= 3

For supplies (2, 1), fees {0.75, 0.875, 1}, λ=3, μ=1, and δ=0.1, the appendix verifies equilibria across the four information regimes.

A.1.2 Equilibrium verification for 𝜆= 3

  • Under OO, the unique equilibrium has lender 2 quote 0.875 and lender 1 quote 1 at both own-supply levels.Source: §A.1.2 · p. 17
  • Under AO, the unique equilibrium has lender 1 quote 0.875 in both competitive states and lender 2 quote 0.875 whenever available.Source: §A.1.2 · p. 18
  • Under OA, the unique equilibrium has lender 1 quote 1 at both own-supply levels and lender 2 quote 0.875 in both competitive states, quoting 1 when alone.Source: §A.1.2 · p. 18
  • Under AA, two equilibria have total payoffs 18.5072 and 19.0564, so the selected payoff is πAA(3)=19.0564.Both quote 1 when alone; their competitive quotes are respectively (0.75, 0.75) throughout, or (0.875, 0.75) in states (1, 1) and (2, 1).Source: §A.1.2 · p. 19

A.1.3 Payoff comparisons

The payoff comparisons contrast asymmetric disclosure regimes in a two-lender example. At the lower arrival rate, AO and OA preserve the same allocations and state transitions but change fees; the higher-rate analysis compares discounted payoffs across regimes.

A.1.3 Payoff comparisons

  • At λ=1, AO and OA give each lender the same allocation probabilities and induce the same transition matrix, so discounted time in every state is equal.The comparison therefore isolates fee differences between the regimes.Source: §A.1.3 · p. 19
  • At λ=1, AO fees exceed OA fees by 1/4 in states (1, 0) and (2, 0), fall by 1/4 in (0, 1), and coincide when both lenders are available.The positive state-time weights for (0, 0) and (2, 0) enter the payoff comparison.Source: §A.1.3 · p. 20
  • Under OO and OA, lender 2 wins whenever both lenders are available; under AO and both AA equilibria, tied competitive fees split the borrower.Both AA equilibria have the same discounted state times as AO and charge 1 when alone.Source: §A.1.3 · pp. 20–21
  • At λ=3 under OA, bookings pay 1 except when both lenders are available, when the fee is 0.875.The fee reduction contributes 3/8 per unit of discounted time in either competitive state.Source: §A.1.3 · p. 21

A.2 Rate bounds and equilibrium verification for Theorem 1

The appendix verifies the theorem's AO and OA equilibria by establishing strict quote incentives and checking complete strategy changes. It uses rate bounds to preserve those incentives when units return.

A.2 Rate bounds and equilibrium verification for Theorem 1

  • The sequence g_j declines, and the chosen fee satisfies α>g_m>β, providing the strict rate bounds used in the quote-incentive arguments.The appendix states that the decline in g_j proves the needed inequality α−β>0.Source: §A.2 · p. 21
  • When μ=0, the verified quote and complete-strategy gains are bounded below by η, with specified complete-strategy gains exceeding βη.The bounds cover lenders' quote choices under AO and OA.Source: §A.2 · pp. 24–25
  • With positive returns, the payoff change for fixed strategies is bounded, allowing the appendix to preserve the verified gains when return rates are introduced.The payoff-gain change at a borrower arrival is bounded by ε_μ; a complete-strategy gain changes by at most 2ε_μ.Source: §A.2 · p. 25
  • Under AO, the specified strategies form the unique pure stationary equilibrium: lender 1 quotes α when competing and 1 alone, while lender 2 chooses α.Lender 2 continues to prefer α against either verified response by lender 1.Source: §A.2 · pp. 23, 26
  • Under OA, lender 1 quotes α at its maximum available supply and 1 at smaller positive supplies, establishing the stated OA equilibrium.The complete-strategy comparison selects α at maximum supply.Source: §A.2 · pp. 24, 26

A.3 Equilibrium verification for Proposition 1

The appendix verifies the AO and OA equilibria for Proposition 1 by checking best responses across the stated parameter region. The resulting payoff comparisons are strict.

A.3 Equilibrium verification for Proposition 1

  • In AO, lender 1 quotes p_L when competing and p_H alone, while lender 2 always quotes p_L.Lender 2 earns more by quoting p_L than p_H against either of lender 1's verified responses.Source: §A.3 · p. 26
  • In OA, the only equilibrium is the third listed strategy profile, with lender 1 quoting (p_H, p_L) at own supplies 1 and 2.Against (p_L, p_L), lender 1 gains by raising its quote at own supply 1.Source: §A.3 · p. 26
  • All 25 payoff comparisons are strict throughout the stated parameter region, establishing the displayed best responses and uniqueness of the AO and OA equilibria.The parameter range includes 13/10(1+d) ≤ r < 11/8(1+d).Source: §A.3 · p. 27

A.4 Gains to both lenders

For the stated example, the appendix computes unique equilibria under AO and OA. Both lenders strictly prefer OA to AO.

A.4 Gains to both lenders

  • Under AO, lender 1 quotes 1 when alone and 0.75 when competing, while lender 2 quotes 0.75 whenever available.This is the unique AO equilibrium for the example.Source: §A.4 · p. 27
  • Under OA, the unique equilibrium has lender 1 quoting 1 whenever available, while lender 2 quotes 0.75 only at state (3, 1).Lender 2 quotes 1 in every other state where it is available.Source: §A.4 · pp. 27–28
  • Both lenders strictly prefer OA to AO in the example with supplies (3, 1), fees {0.75, 1}, λ=2, μ=1, and δ=0.1.The appendix reports that the exact payoff differences between the equilibria satisfy the strict preference.Source: §A.4 · pp. 27–28

B Ownership Fragmentation: Equilibrium Verification

The appendix verifies that Proposition 3’s strategies are the unique pure stationary equilibrium in each game throughout the stated parameter region.

B Ownership Fragmentation: Equilibrium Verification

  • The proposed strategies are the unique pure stationary equilibrium in each game throughout the stated parameter region.The appendix establishes this by showing the proposed strategies are mutual best responses and no other strategy combination is an equilibrium.Source: B Ownership Fragmentation: Equilibrium… · pp. 28–29
  • The verification enumerates 64 remaining strategy combinations in the (2, 2) game under OA and 256 in the (2, 1, 1) game under OAA.In OAA, the two informed lenders need not use identical strategies.Source: B Ownership Fragmentation: Equilibrium… · pp. 28–29
  • The stated parameter region is 0 < d ≤ 1, 4 + 4d ≤ r ≤ 21/5 + 4d, and 899/1000 ≤ α ≤ 901/1000.Payoff differences are rational in r and d and affine in α, allowing endpoint checks once denominators are shown positive.Source: B Ownership Fragmentation: Equilibrium… · pp. 28–29
  • Exact arithmetic verifies 290 payoff differences through 1160 polynomial checks, with every required payoff difference strictly positive throughout the parameter region.The checks show each informed lender’s proposed strategy is its unique best response to every combination of rivals’ remaining strategies, followed by the corresponding check for lender 1.Source: B Ownership Fragmentation: Equilibrium… · p. 29
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