number of vessels who fish in the same location (<sup>_∂πi,l_</sup> _∂I_<sup><u>(</u></sup><sup>_<u>μ</u>_</sup> _−_<sup>_l_</sup> _i,l_<sup>_<u>,I−i,l</u>_</sup><sup><u>)</u></sup> _<_ 0), expected profit from fishing in location _h_ is higher than in _g_ (E[ _i, h_<sup>_∗_</sup> ] _>_ E[ _i, g_<sup>_∗_</sup> ] _∀i_ ), because marginal congestion costs are the same in _h_ and _g_ and the number of vessels who choose _h_ is not higher by assumption ( _Ih_<sup>_∗≤I_</sup> _g_<sup>_∗_).Becausethetotalnumberofvesselsisfixed,thenumberofvessels</sup> who choose location _k_ when _C_ = 1 is greater than or equal to the number of vessels who choose location _k_ when _C_ = 0 ( _Ik_<sup>_∗_</sup> _|C_ =1<sup>_≥I_</sup> _k_<sup>_∗_</sup> _|C_ =0<sup>).Sincevesselshavethesamebeliefsabout</sup> _k_ ’s productivity in both states of the world, expected profit from fishing in location _k_ when _C_ = 0 is at least as great as expected profit from fishing in _k_ when _C_ = 1 (E[ _i, k_<sup>_∗_</sup> _|C_ = 0] _≥_ E[ _i, k_<sup>_∗_</sup> _|C_ = 1] _∀i_ ). Since ( _Ig_<sup>_∗, I_</sup> _k_<sup>_∗_)isaBayes-Nashequilibrium,E[</sup><sup>_i, g∗_]=E[</sup><sup>_i, k∗|C_=0]by</sup> the proof of Proposition 2a. Then E[ _i, h_<sup>_∗_</sup> ] _>_ E[ _i, k_<sup>_∗_</sup> _|C_ = 1] _∀i_ because E[ _i, h_<sup>_∗_</sup> ] _>_ E[ _i, g_<sup>_∗_</sup> ] = E[ _i, k_<sup>_∗_</sup> _|C_ = 0] _≥_ E[ _i, k_<sup>_∗_</sup> _|C_ = 1]. Then ( _Ih_<sup>_∗, I_</sup> _k_<sup>_∗_)cannotbeaBayes-Nashequilibriumbythe</sup> _<u>∂Il</u>_ proof of Proposition 2a. Contradiction. Therefore, _∂μ_ ̃ _l_<sup>_>_0whencongestioncostsarethe</sup> same across locations.

However, marginal congestion costs are in fact higher in location _h_ (because _h_ covers less area than _g_ and _k_ ). Though vessels believe mean productivity is higher in _h_ than in _g_ and _k_ , the higher marginal congestion cost in _h_ counteracts the effect of higher mean productivity on the number of vessels who choose _h_ . For this reason, it is not necessarily the case that the closures policy increases fishing near closures ( _Ih_<sup>_∗> I_</sup> _g_<sup>_∗_).</sup> To see how higher marginal congestion costs in _h_ reduce the number of vessels who choose ̃ ̃ _h_ , consider the case when _μh_ = _μg_ and suppose the contradiction that _Ih_<sup>_∗≥I_</sup> _g_<sup>_∗_.Expected</sup> profits are lower in _h_ than in _g_ because marginal congestion costs are higher in _h_ (E[ _i, h_<sup>_∗_</sup> ] _<_ E[ _i, g_<sup>_∗_</sup> ] _∀i_ ). Since _μ_ ̃ _k_ is the same in both states of the world and _Ik_<sup>_∗_</sup> _|C_ =0<sup>_≥I_</sup> _k_<sup>_∗_</sup> _|C_ =1<sup>(because</sup> _Ig_<sup>_∗≤I_</sup> _h_<sup>_∗_),expectedprofitin</sup><sup>_k_when</sup><sup>_C_= 0islessthanorequaltoexpectedprofitin</sup><sup>_k_when</sup> _C_ = 1 (E[ _i, k_<sup>_∗_</sup> _|C_ = 0] _≤_ E [ _i, k_<sup>_∗_</sup> _|C_ = 1] _∀i_ ). Since ( _Ig_<sup>_∗, I_</sup> _k_<sup>_∗_)isaBayes-Nashequilibrium,</sup> E[ _i, g_<sup>_∗_</sup> ] = E[ _i, k_<sup>_∗_</sup> _|C_ = 0] by the proof of Proposition 2a. Then E[ _i, h_<sup>_∗_</sup> ] _<_ E[ _i, k|C_ = 1] _∀i_ because E[ _i, h_<sup>_∗_</sup> ] _<_ E[ _i, g_<sup>_∗_</sup> ] = E[ _i, k|C_ = 0] _≤_ E[ _i, k|C_ = 1]. Then ( _Ih_<sup>_∗, I_</sup> _k_<sup>_∗_)cannotbeaBayes-</sup> Nash equilibrium by the proof of Proposition 2a. Contradiction. Therefore, the higher

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