## Abstract

We consider an *externality game* in which nonatomic agents choose from a finite set of alternatives and disutility is determined only by the number of agents choosing each alternative. The equilibrium is defined with respect to the agents’ choices so that taste heterogeneity, modeled through randomized parameters, can be estimated from collective choice data. The joint density of the taste parameters is computed by a biquadratic inverse optimization process that matches observed choices to the equilibrium condition associated with a set-valued best-response function, and hence imposes no prior assumptions on the taste parameters. The model is capacitated with disutility generalized into an arbitrary function that is continuous in attributes and measurable in taste parameters. The existence of an equilibrium under such a disutility model is established by observing that, in an externality game, the proposed aggregate equilibrium is actually equivalent to the agent-specific Nash equilibrium previously established by Schmeidler (J Stat Phys 7(4):295–300, 1973). In a comparison test on intensive metro route-choice data, we demonstrate that the proposed model is a good alternative to existing nongame choice models. An extended test also demonstrates the advantage of the general disutility model in describing agents choice behaviors in other contexts.

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## Acknowledgements

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2021R1A2C1011341). The authors give their thanks to Eric Hong for his translation and proofreading. The authors are also grateful to Max Kapur for his proofreading and helpful suggestions for revision of the paper.

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## A BNN and best-response dynamics

### A BNN and best-response dynamics

For current aggregate choice *x* \(\in \) *X*, the \(x_i\) of each alternative *i* \(\in \) *N* can be disaggregated into \(x_i(\gamma )\), the portions from \(\gamma \in {\hat{\Gamma }}\): \(x_i\) \(=\) \(\sum _{\gamma \in {\hat{\Gamma }}} x_i(\gamma )\), *i* \(\in \) *N*. We define a response function \(r_i(x; \gamma )\) and step size \(\lambda (x; \gamma )\) so that the next point \(x'_i\) is defined as \(x'_i(\gamma )\) \(=\) \(x_i(\gamma )\) \(+\) \(\lambda (x;\gamma ) \left( r_i(x; \gamma ) - x_i(\gamma )\right) \). The Brown-von Neumann-Nash (BNN) dynamics use \(r_i(x;\gamma ) = {\hat{g}}(\gamma ) \frac{p_i(x; \gamma )}{\sum _{j \in N} p_j(x; \gamma )}\), and \(\lambda (x ; \gamma ) = \frac{\sum _{j \in N} p_j(x;\gamma )}{1+\sum _{j \in N} p_j(x;\gamma )}\), where \(p_i(x ; \gamma ) = \max \left\{ \frac{1}{{\hat{g}}(\gamma )}\sum _{j \in N} x_j(\gamma ) u_j(x ; \gamma ) -u_i(x; \gamma ), 0\right\} \). On the other hand, the best-response dynamics consider the set \(S(x;\gamma )\) of alternatives with the minimum disutility for \(\gamma \). For simplicity, write *S* \(\equiv \) \(S(x;\gamma )\) hereafter. Let \(r_i(x; \gamma )\) \(=\) \({\tilde{\rho }}_i(\gamma )\) if \(i \in S\) or 0 otherwise, where \({\tilde{\rho }}(\gamma )\) is generated randomly from the simplices \(\{\rho (\gamma ): \sum _{i \in S} \rho _i(\gamma ) = {\hat{g}}(\gamma )\), \(\rho _i(\gamma ) \ge 0\}\) for \(\gamma \) \(\in \) \({\hat{\Gamma }}\). We adopted the usual moderated step-size rule \(\lambda _k = \frac{1}{k+1}\) (see e.g. Fisk (1980)). In both dynamics, we hired stopping criterion \(\Vert x- x'\Vert \) < \(\epsilon \Vert x\Vert \) for \(\epsilon = 0.01\).

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Hong, SP., Kim, K.M. & Ko, SJ. Estimating heterogeneous agent preferences by inverse optimization in a randomized nonatomic game.
*Ann Oper Res* **307, **207–228 (2021). https://doi.org/10.1007/s10479-021-04270-2

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### Keywords

- Nonatomic externality game
- Taste heterogeneity
- Taste-parameter randomization
- Biquadratic inverse optimization