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:

• Rasmus Bååth
• Darren Wilkinson

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.