Chinese Restaurant Game - Part I: Theory of Learning with Negative Network Externality

In a social network, agents are intelligent and have the capability to make decisions to maximize their utilities. They can either make wise decisions by taking advantages of other agents’ experiences through learning, or make decisions earlier to avoid competitions from huge crowds. Both these two effects, social learning and negative network externality, play important roles in the decision process of an agent. While there are existing works on either social learning or negative network externality, a general study on considering both these two contradictory effects is still limited. We find that the Chinese restaurant process, a popular random process, provides a well-defined structure to model the decision process of an agent under these two effects. By introducing the strategic behavior into the non-strategic Chinese restaurant process, in Part I of this two-part paper, we propose a new game, called Chinese Restaurant Game, to formulate the social learning problem with negative network externality. Through analyzing the proposed Chinese restaurant game, we derive the optimal strategy of each agent and provide a recursive method to achieve the optimal strategy. How social learning and negative network externality influence each other under various settings is also studied through simulations.

💡 Research Summary

This paper addresses a gap in the literature on decision‑making in networks where agents must simultaneously contend with two opposing forces: the desire to learn from the actions and signals of others (social learning) and the need to avoid crowding effects that reduce individual payoff (negative network externality). Existing works typically treat these phenomena in isolation—social learning models assume that later agents’ utilities are unaffected by earlier agents’ choices, while network‑externality models (often positive) ignore learning dynamics. To bridge this divide, the authors introduce the “Chinese Restaurant Game” (CRG), a strategic extension of the non‑strategic Chinese Restaurant Process (CRP), a well‑known construct in Bayesian non‑parametrics.

In the CRG, there are K tables (actions) and N customers (agents) arriving sequentially. The size of each table, Rj(θ), depends on an underlying state θ (e.g., which physical table is large or small). θ is unknown to the agents; they receive noisy signals (advertisements, reviews) and can observe previous agents’ table selections. An agent i’s utility is U(Rxi, ni), where xi is the chosen table and ni is the number of agents already seated there. U is increasing in table size and decreasing in ni, thereby encoding a negative externality: the more agents share a table, the lower each individual’s payoff.

The analysis proceeds in four stages. First, with perfect information (θ known) and simultaneous moves, the authors derive the Nash equilibrium: agents distribute themselves so that the expected utility of each occupied table is equal, reflecting a natural load‑balancing outcome under negative externality. Second, they consider a sequential game with perfect information; here the early mover enjoys a “first‑mover advantage” because his choice serves as a public signal that influences later agents’ beliefs and actions. The equilibrium combines a “lead‑and‑avoid” pattern where early agents may deliberately select less crowded tables to preserve future utility.

The core contribution lies in the third stage, where θ is uncertain. Agents perform Bayesian updating using both exogenous signals and the observed actions of predecessors. The authors formulate a recursive dynamic programming solution: at each decision epoch i, agent i computes the posterior distribution πi(θ) given all prior information, evaluates the expected utility for each table x as Eπi