Safe coin bias when the opponent chooses second

Alice and Bob each choose the heads probability of their own coin, then both toss. If both show heads, Alice pays Bob $6; if both show tails, Alice pays Bob $4; if they differ, Bob pays Alice $5. Alice must announce her probability $p_1$ before Bob picks his $p_2$. The set of $p_1$ for which Bob's expected profit is at most $0$ whatever $p_2$ he picks is an interval $[a,b]$ with rational endpoints. Find $b-a$.

$1/99$. Bob's expected profit simplifies to $(20p_1-9)p_2+(4-9p_1)$, which is linear in $p_2$, so he picks $p_2=0$ or $1$. Requiring both $4-9p_1\le0$ and $11p_1-5\le0$ gives $p_1\in\left[\tfrac49,\tfrac5{11}\right]$, of width $\tfrac{1}{99}$.