How Much Should You Bet In A Game That Favors You ?
Here is an interesting problem. I’m going to lay out a favorable bet. You will have the opportunity to place a bet multiple times, and resolve each one before moving onto the next one. How much should you bet each time so that the you whose luck is exactly average will win the most money ?
Here is the gambling scenario –
- You start with 100 dollars.
- We will flip a fair coin, and if it is heads, you win.
- Before each flip you get to choose how much money you want to bet on that flip
- If you win, you double the amount of money that you bet. If you lose you lose half of the money that you bet. e. if you bet the full 100 dollars you started with, and you win you have 200 dollars, and if you lose you have 50 dollars.
- You can play this game up to 100 times.
Now clearly this is a favorable bet. On the first bet, if you bet 100 dollars, you have a 50/50 chance of winning vs losing. 50 percent of 200 dollars plus 50 percent of 50 dollars means the expected value of that bet is 125 dollars, for a gain of 25 dollars.
However just as clearly if you place that bet 100 times you are just as likely to win as many bets as you are to lose those bets. So if you bet all your money each time, then times that you get lucky and win more times than you lose, for instance win 56 times and lose 44 times, you will win a lot of money. However the person whose luck is exactly average will win 50 times and double their money, and lose 50 times and cut their money in half, and leave with the same 100 dollars they started with.
We want that average lucky person to win the most money possible, so how much should you bet ?
To simulate that I ran a python program through a couple million people betting different percentages of their bankroll. You can download that python program, the one that generated the charts below, and the associated Excel file here. This chart shows how much money the 50th percentile person would have depending on how much they bet as a percentage of their bankroll.
So if you knew that you were going to win exactly 50% of your bets, your optimal bet amount exactly half of your bankroll each time.
Now if you think that you are going to be lucky, and win more than average, or you want to maximize your potential winnings you could bet more. For a 90th percentile person, you are likely to win 56 bets out of the total 100. That is a lot better than only winning 50 bets, but it is still a lot of losses. This is what the winning curve looks like for the 90th percentile person
Even though you were in the 90th percentile of luck, betting all your money each time would be a mistake. If you knew that you were going to win exactly 56 of your 100 bets, your optimal bet size would be 68% of your bankroll each time. So betting more money on a positive expected value game can increase your net expected value, but it would come at an increased risk if you don’t run lucky.
Even Winnings – In A Game That Favors You
The scenario outlined above may not be the most likely scenario you will encounter. I have given that scenario some thought because it was in a video game that I played when I was younger. At the end of each level you were given the opportunity to make a bet 3 times and double or halve the number of coins you had collected in the level each time. ( Bonus points if you know what Gameboy game I’m referring to) There are situations in business or with stock option investing that could replicate that, and you could either double your money or lose half your money, but if you are placing bets at a casino the scenario you are most likely to encounter is that you will win or lose the same amount of money. However your odds of winning and losing may not be 50/50
Let’s say that you had a game where you would win or lose your bet, but that the odds of winning were 55% and you only had a 45% chance of losing. i.e. imagine that you were betting on red or black on a game of roulette, except that the color distribution favored you instead of the casino. You have the opportunity place this bet 100 times.
Here betting 100% of your money is obviously a bad idea, because when you lose you will lose all of your money, and you will lose almost half the time. The curve of winnings vs bet rate looks like this for the 50th percentile lucky person
As it turns out, the best amount to bet here is 10% of your bankroll each time.
That 10% number is suspiciously the same as the amount of advantage that you have in the game, i.e. the difference between 55% win rate and 45% loss rate.
If we run the simulation again with a 60% win rate, and a 65% win rate
We see that the best bet amount for 60% is to bet 20% of your bankroll, and the best bet amount for a 65% win rate is to bet 30% of your bankroll. It turns out that the best bet is exactly the advantage that you have in the game.
It turns out that this optimal amount to bet was first discovered in 1956, and is known as the “Kelly Criterion”. This was named after John Larry Kelly, who was an associate of the famous Claude Shannon, who went on to use this formula as well as other mathematical techniques to make a substantial amount of money in the casinos and in Wall Street.
The Kelley Criterion equation is
- f* is the percentage of the current bankroll to bet
- p is the probability of winning the bet
- q is the probability of losing the bet ( 1 – p )
- b is the net odds received on the bet. e. “b to 1”. So for a flip of the coin, where you bet 1 and get 2 if you win (your original 1 + 1) b would be 1. If you are betting 1 dollar and get 6 if you win, i.e. 5 to 1 odds, b is 5
This turns out to match our results exactly. For the first example where we got either double or half, b is 2, and p and q are both .5. The equation becomes
And f* is calculated to be .25 = 25% of our bankroll as the optimal bet. Now the graph indicated that we should be betting 50% of the bankroll. However since we could only lose half of that 50% of the bankroll in the double or half bet, that is equivalent to betting 25% of the bankroll when using the same terminology as the Kelly Criterion.
For the bets where our payoff is the same as our bet, and our odds of winning are either 55%, 60%, or 65%, b is 1, and an example equation becomes
Which shows that we should be betting 20% of the bankroll for the 60% chance of winning case, which is exactly the same as the python code determined.
For most real world scenarios, where you are betting against the house which has a house edge, f* becomes negative, which means that you shouldn’t be playing that game. Truthfully it means that you should take the other side of the wager, become the house, and make them bet against you!