Range of prices in horse market:-

Show that the requirement that the number of owners who sell their horses must equal the number of nonowners who buy horses, together with the arguments above, implies that the common trading price p∗ is at least σk∗, at least β_k∗+1, at most β_k∗ , and at most σ_k∗+1. That is, p∗ ≥ max{σ_k∗ , β_k∗+1} and p^∗ ≤ min{β_k∗ , σ_k∗+1}

Finally, I argue that in any action in the core a player whose valuation is equal to p∗ trades. Suppose non  owner i's valuation is equal to p∗. Then owner i's valuation is less than p∗ and owner i + 1's valuation is greater than p∗ (given my assumption that no two players have the same valuation), so that exactly i owners trade. Thus exactly i non owners must trade, implying that non owner i trades. Symmetrically, a owner whose valuation is equal to p∗ trades. In summary, in every action in the core of a horse trading game,

• every non owner pays the same price for a horse
• the common price is at least max{σ_k∗, β_k∗+1} and at most min{β_k∗, σ_k∗+1}
• every owner whose valuation is at most the price trades her horse
• every non owner whose valuation is at least the price obtains a horse.

The action satisfying these conditions for the price p∗ yields the payoffs.