Two-Sided Market Making With Overround
You are making a two-sided market on a binary event. The true probability that the event occurs is $p$. You quote decimal odds $O_{\text{yes}}$ and $O_{\text{no}}$ (payout includes stake, so a customer who bets $\
$ on "yes" receives $O_{\text{yes}}$ if the event occurs and loses the dollar otherwise).
A customer arrives with a private belief $p_c$ about the event probability, drawn uniformly on $[0, 1]$. The customer computes their expected value on each side:
- Bet "yes": $\text{EV} = p_c \cdot O_{\text{yes}} - 1$
- Bet "no": $\text{EV} = (1 - p_c) \cdot O_{\text{no}} - 1$
If exactly one side has positive expected value, the customer takes that side. If both sides have positive expected value, the customer takes the side with higher EV. If neither side has positive EV, the customer does not trade. The customer always stakes $\
$.
1. Derive your expected profit per arriving customer as a function of $(O_{\text{yes}}, O_{\text{no}})$ and $p$.
2. Find the pair of odds $(O_{\text{yes}}^{*}, O_{\text{no}}^{*})$ that maximizes your expected profit, and verify that the maximum is strictly positive for all $p \in (0,1)$.
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