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- Property 1 Documents de travail du Centre d'Economie de la Sorbonne - 2016.37 • If the players at a or d prefer an interior location it is given by U[b,c] • If the player at b has an interior best reply, it is given by the uniform random variable U[a,c] or U[c,d] • If the player at c has an interior best reply, it is given by the uniform random variable U[b,d] or U[a,b] • If the best reply of the player at b is U[c,d], the best reply of the player at c is U[b,d] • If the best reply of the player at c is U[a,b], the best reply of the player at a is U[a,c] Lemma 12 Suppose that U ∈ {U[a,c], U[b,c], U[b,d], U[c,d], U[a,b]} is an interior best reply of some player, then P(d(U, M) < N ) ≥ c2, where c2 > 0 is a constant independent of N. Proof: For uniform variables on the intervals [a, c], [b, c] and [b, d], the result is obvious since these intervals contain M. For example [a, c] = [a, M] S
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