bcf: Bayesian Coin Flip
The bcf package simulates turn-based head-to-head competitions by approximating such games by simple coin flips. Each player or team is modeled as a coin with some probability distribution for landing heads-up in a given flip. The player or team that obtains a Heads result first wins, and ties are broken by running sub-games among the participating players as necessary.
Players' or teams' posterior distributions are obtained via Approximate Bayesian Computation (abc_coin_flip_game()). See:
An example game:
| Player 1 | Player 2 | Player 3 | Result |
|---|---|---|---|
| Heads | Heads | Heads | All tied; re-flip. |
| Heads | Tails | Tails | P1 wins; P2 & P3 re-flip. |
| — | Heads | Heads | P2 & P3 re-flip. |
| — | Tails | Heads | P3 finishes 2nd; P2 finishes 3rd. |
Presently, bcf imposes a beta distribution on each player, both initially and after each game. This is an approximation; the result of a multi-way coin flip, allowing for ties, doesn't seem to result in the usual binomial or Bernoulli likelihood function that would be conjugate to a beta distribution.
Usage
bcf provides one method for creating new players, another for simulating the coin flip game, and a third for pushing the result of a game onto a participating player. The package also provides S3 methods for printing and plotting (both players and games) and an as.data.frame method on games.
library(bcf)
tom <- new_player("Tom", alpha = 1.2, beta = 1.2)
bob <- new_player("Bob", alpha = 1.2, beta = 1.2)
print(tom)UUID: bab835b2-c361-11e7-9906-f45c899c4b7b
Name: Tom
Games: 0
Wins: 0
Losses: 0
Est. Distribution: Beta(1.200, 1.200)
MAP Win Percentage: 50.000
g1 <- abc_coin_flip_game(list(tom, bob), result = c(1, 2), iters = 1e5, cores = 4L)No. players: 2
Assign result: 1, 2
Iters: 1e+05
CPU cores: 4
Workloads: 25000, 25000, 25000, 25000
print(g1)# A tibble: 2 x 5
Tom Bob outcome n pct
<dbl> <dbl> <chr> <int> <dbl>
1 1 2 *** 49981 50
2 2 1 50019 50
tom <- update_player(tom, g1)
print(tom)UUID: 0f60b9a4-c362-11e7-9906-f45c899c4b7b
Name: Tom
Games: 1
Wins: 1
Losses: 0
Est. Distribution: Beta(1.872, 1.145)
MAP Win Percentage: 85.707
To-Do
There's still a long list of ways to make bcf better:
- Fit a density to each player, not necessarily a beta distribution.
- Sample directly from the posterior, not the joint distribution.
- Compute the likelihood for N-player games.
- Infer expected win/loss records from a player's win distribution.
- Compute and display a reasonable uncertainty on a player's win probability.
- Drop the lazy dplyr import.
Contact
Find me at tshafer.com or on Twitter at @tomshafer.