Why not just ban the hero with the best win rate? A win rate mixes the hero with the players who choose it, a ban changes what both
teams draft, and the other team answers your bans without showing who they are. So Ban Call plays the ban phase out as a game with hidden types and solves
it backwards.
One lobby, three possible types
Illustrative numbers. Hover anything for its arithmetic.
Backward induction
Click a term to find it in the figure.
After six bans
V6(s6) = 𝔼θ | s6[1/282 Σi,j P(WA = 1 | LAi, LBj, m)]
The average over 784 lineup pairings, then over the types.
Your turn
Vk(sk) = maxb ∈ 𝓛(sk)V̄k+1(skb)
The best legal ban in expectation, one per information set.
Their turn
Vk(sk) = 𝔼θ | sk[Σb σ̂B(b | sk, θB) Vk+1(skb)]
Their likely bans from a fitted model, not their worst.
Choosing a ban is harder than it looks. A ban removes a hero from both teams, it changes what both teams draft, and you usually can't see who is on the
other side. Ban Call plays your lobby out many times before you choose. The two figures show how, with the model's own numbers for the lobby set above: loading.
Forward, then back
One simulated game, marked with the first ban it tried
Inside one ending
Where a branch ends, all six bans are in. The seats you can't see are filled with real players from your rank, everyone picks, and every lineup is scored
against every opposing one. This repeats at every ending.