Kelly Criterion, Should you count Tie?

Assume a baccarat promotion that if you lose, you only lose 80% of the bet.
You always bet Banker.

Probability counting Tie:
Banker wins 0.4586%
Player wins 0.4462%
Tie 0.0952%
(.95*.4586-.8*.4462)/.95=0.0829
If counting Tie, the optimal kelly bet 8.29% of the bankroll.

Probability not counting Tie:
Banker wins 0.5068%
Player wins 0.4932%
(.95*.5068-.8*.4932)/.95=0.0915
If not counting Tie, the optimal kelly bet is 9.15% of the bankroll.

Which is correct? Should you count Tie?

Quote: ReyGarcia

Assume a baccarat promotion that if you lose, you only lose 80% of the bet.
You always bet Banker.

Probability counting Tie:
Banker wins 0.4586%
Player wins 0.4462%
Tie 0.0952%
(.95*.4586-.8*.4462)/.95=0.0829
If counting Tie, the optimal kelly bet 8.29% of the bankroll.

Probability not counting Tie:
Banker wins 0.5068%
Player wins 0.4932%
(.95*.5068-.8*.4932)/.95=0.0915
If not counting Tie, the optimal kelly bet is 9.15% of the bankroll.

Which is correct? Should you count Tie?

We can do it either way, but we should be sure to use the correct variance for each case.

With tie:

EV = 0.95 * 0.4586 - 0.8 * 0.4462 + 0 * 0.0952 = 0.0787

Variance = 0.4586(0.95 - 0.0787)² + 0.4462(-0.8 - 0.0787)² + 0.0952(0 - 0.0787)² = 0.6933

Kelly fraction = 0.0787 / 0.6933 = 11.4%



Without tie:

EV = 0.95 * 0.5069 - 0.8 * 0.4931 = 0.0870

Variance = 0.5069(0.95 - 0.0870)² + 0.4931(-0.8 - 0.0870)² = 0.7655

Kelly fraction = 0.0870 / 0.7655 = 11.4%

Quote: ReyGarcia

Assume a baccarat promotion that if you lose, you only lose 80% of the bet.
You always bet Banker.

Probability counting Tie:
Banker wins 0.4586%
Player wins 0.4462%
Tie 0.0952%
(.95*.4586-.8*.4462)/.95=0.0829
If counting Tie, the optimal kelly bet 8.29% of the bankroll.

Probability not counting Tie:
Banker wins 0.5068%
Player wins 0.4932%
(.95*.5068-.8*.4932)/.95=0.0915
If not counting Tie, the optimal kelly bet is 9.15% of the bankroll.

Which is correct? Should you count Tie?

By the way, dividing the EV by the variance is only an approximation for the Kelly fraction.

An exact way to get the Kelly fraction, f, for your game is to maximize the following function:

( 1 + 0.95*f )^banker( 1 - 0.80*f )^player

The probabilities of a banker win and a player win are denoted above by "banker" and "player," respectively.

Using the above method with the combinations for banker and player for an 8-deck game from this Wizard's page gives a Kelly fraction of 11.43993% for your game. (It gives the same answer with or without considering ties.)

The EV/variance method gives 11.347% with ties and 11.358% without ties.

Thank you for the detailed explanation.
I think I got it all wrong, I failed to consider variance.
I copied kelly formula from Wikipedia, "divide EV by the amount of win". Does it only work when only two outcomes are involved, one is winning a certain amount, one is losing the entire bet(not losing 80% in my example)?