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Introduction
In many businesses, independent firms engage in capacity sharing. Airline companies generally sign code-share agreements with partners to share flight seats ( $\mathrm{Hu}$ et al. 2007), and similarly, cargo carriers set up agreements to provide vessel capacity for each other to use (Reuters 2008). In the hospitality industry, it is often observed that nearby hotels share rooms with each other to accommodate an unexpected overflow of guests, which can result from the hotels’ overbooking policies and lastminute walk-ins (Toh and Dekay 2002, Vora 2019). Public sectors also see collaborative capacity sharing. For example, fire and police departments in close communities establish mutual aid agreements to
share 911 call centers for better response to emergencies (Austin Monitor 2019). Hospitals share expensive diagnostic equipment, laboratory facilities, intensive care units, specialists, and surgeons in case of emergency, and they operate ambulance diversion among emergency departments (EDs) (Deo and Gurvich 2011, Tuller 2016). Such types of collaborations were especially important during the coronavirus disease 2019 (COVID19) pandemic as COVID-19 cases tended to be clustered. Although some regions’ healthcare resources experienced a patient surge, other regions had idle capacity (CDC 2020).

A key challenge for businesses to establish and maintain their collaboration of capacity sharing is how to

align the incentives of independent firms. The operations management literature has tackled this challenge by studying efficient and incentive-compatible monetary-payment contracts between firms. Applications include, for example, revenue sharing in airline alliances (Wright et al. 2010), coordination of call center outsourcing (Gans and Zhou 2007), and inventory transshipment (Rudi et al. 2001), to name a few.

However, in some other cases, monetary incentive based on transfer payments may not be an appealing solution. For example, legal and bureaucratic issues can make it undesirable to set up monetary transfers agreements between public service providers, such as public hospitals and fire and police departments, especially when they involve separate local communities and governments. Private sector firms may be subject to antitrust regulations when they engage in monetary transactions (DOJ 1995) or information exchange (Stein and Li 2021). Furthermore, it is generally costly to negotiate and enforce transferpayment contracts. This is especially true if the transfer prices depend on parameters of the market environmentsuch as demand, service rates, pricing strategies, and so on-and need to be agreed upon in advance when firms have high uncertainties. It is known that no matter how small negotiation costs are, they can dramatically change the outcome of a negotiation process, and in some cases, no collaboration is the only possible equilibrium regardless of the bargaining formats (Anderlini and Felli 2001).
In this paper, we investigate the question of how to support and sustain capacity-sharing collaboration between service providers without monetary transfer payments. This type of collaboration relies on reciprocity as the incentive mechanism. Field and experimental studies by psychologists, anthropologists, and behavioral economists, such as those in Fiske (1992) and Berg et al. (1995), have demonstrated that people tend to return a favor after receiving one.

Our goal is to identify operational conditions that would allow for reciprocal capacity sharing without transfer payments and to measure the efficiency (or inefficiency) of such collaboration relative to two benchmarks (i.e., a transfer-payment agreement that implements central coordination and an agreement based on perfect monitoring). We propose a parsimonious game theory framework, in which two firms dynamically choose whether to accept each other’s customers. We capture the reality that firms’ real-time capacity utilization is often private information. In such a case, trigger strategies for repeated games with perfect monitoring-namely, firms terminate the collaboration agreement when either of them ever declines to share capacity upon request-do not work because a firm can claim that no capacity is available to share at the time the request is made.

We model two firms’ interactions by a continuoustime repeated game with imperfect monitoring and focus on a class of public strategy, in which the firms’
real-time capacity-sharing decision depends on an intuitive and easy-to-implement accounting device, namely the current net number of transferred customers. We refer to such an equilibrium as a trading-favors equilibrium (TFE). We characterize the condition in which capacity sharing takes place in such an equilibrium; in particular, it suggests that the firms engage in trading favors when both have moderate traffic intensity (i.e., the ratio of demand and capacity rates). We demonstrate that some degree of efficiency loss, as compared with a central planner’s solution, is necessary to induce reciprocity because if one firm commits to always sharing capacity, then the other sees no harm in deviating from that. Our numerical studies find that (1) the efficiency loss is small when the two firms have similar traffic intensity, even if they are different in service-capacity scale; (2) the efficiency loss is considerably large when the two firms have significantly different traffic intensity; and (3) the trading-favors mechanism, surprisingly, can outperform the perfect-monitoring benchmark when the two firms exhibit high asymmetry in terms of servicecapacity scale or traffic intensity as the smaller firm tends to deviate from collaboration.

In the remainder of the paper, we review the related literature in Section 2, introduce our model setup in Section 3 , define and analyze the trading-favors equilibria in Section 4, introduce the two benchmarks and present numerical studies in Section 5, and conclude our paper with a discussion of managerial implications in Section 6.

Literature Review
2.1. Economics Literature on Trading Favors
Economists have conducted studies that model firms’ reciprocal relationship by repeated games with imperfect monitoring. Möbius (2001) proposes the first model of trading favors; he considers two agents who encounter opportunities to do each other costly favors over time. The time epochs at which these opportunities arrive are privately observed by the agents and driven by independent Poisson processes. The paper considers a simple class of perfect public equilibrium, in which the agents’ choices depend only on the net number of favors that they have traded, and the paper shows that, in equilibrium, agents are willing to offer favors at every opportunity as long as the absolute value of the net number of favors is below a threshold. The paper characterizes the maximum value of such a threshold. Several papers extend the model of Möbius (2001). Hauser and Hopenhayn (2005) allow favors to be divisible and appreciate/depreciate over time, whereas Kalla (2010) considers the case where the agents are risk averse and have discount factors that are private knowledge. Lau (2013) considers the case where the payoffs associated with favors are random. Different from those trading-