Choosing a One-Off Coin Flip Bet with Kelly and Utility
Summary
The document compares ways to size a wager in a one-time, even-odds coin-flip game with asymmetric payouts. Applying the Kelly criterion gives a quarter of bankroll for the stated payoff, which maximizes long-run logarithmic growth when bets can be repeated. A separate expected-utility approach evaluates the wealth outcomes from winning or losing and chooses the stake that maximizes their average utility. With logarithmic utility, it produces the same fraction as Kelly.
The answers emphasize that the result depends on the objective: long-run growth and one-shot utility are different decision frameworks, and other utility functions can imply other stakes. One response recommends betting nothing because the game offers no edge. The discussion does not establish a universally correct wager; it depends on the payoff interpretation, the bettor’s wealth and preferences, and whether the opportunity is genuinely repeatable.
Key ideas
- Kelly sizing selects a bankroll fraction to maximize long-run growth under repeated betting.
- For the stated even-odds payout, Kelly implies wagering one quarter of bankroll.
- Expected utility can size a one-time bet according to the bettor’s preferences over wealth outcomes.
- Log utility yields the same stake as Kelly in this example, while other utility functions can differ.
- A bettor who sees no edge may rationally decline the wager.
Tags
Full text
# Trading interview gambling question
# Trading interview gambling question
You're invited to a one-on-one coin-flip gambling game. Your opponent has 1 million USD on hand (the max you can bet is 1 million USD). The payouts for flipping heads and tails are as follows:
This game is a one-off gamble; no second chances. How much would you bet?
(This was from a trading interview so it likely involves choosing the right risk-reward profile. Is there a right answer? Or is this a subjective question?)
## Answer by RRG (score 13, accepted)
https://quant.stackexchange.com/a/36127
The Kelly criterion gives the fraction, $f$, of the current bankroll to bet in order to maximize the longterm growth. The criterion is given by $$ f = \frac{bp-q}{b}, $$ where $b$ is the winnings received on \$1 bet, $p$ is the probability of winning, and $q=1-p$ is the probability of losing the bet of \$1.
In your case $b=2$, $p=q=0.5$ so the optimal fraction to bet is $$ f = \frac{2\cdot0.5-0.5}{2} = 0.25. $$ That is 25% of your bankroll or \$250k.
## Answer by Chris Taylor (score 18)
https://quant.stackexchange.com/a/36140
An alternative approach is to size your bet to maximize your expected utility, which is assumed to be given by a function $u(w)$ of your total wealth $w$. This could be a better approach than using the Kelly criterion, because the Kelly fraction gives the amount to bet if you want to maximize your long-term growth rate, assuming that you will bet a large number of times, but in this case you are told that you only get one chance to bet.
If you bet a fraction $x$ of your bankroll, you will have $1+2x$ if you win and $1-x$ if you lose, so your expected utility is
$$ \tfrac{1}{2}u(1 + 2x) + \tfrac{1}{2}u(1 - x) $$
Maximizing this is equivalent to maximizing $u(1+2x) + u(1-x)$. In the special case of log utility $u(w)=\log w$ you require
$$ \frac{d}{dx} \left( \log(1+2x) + \log(1-x) \right) = \frac{2}{1+2x} - \frac{1}{1-x} = 0 $$
which you can solve to give $x = 1/4$, the same answer as if you used Kelly betting to maximize your long-term growth. Other utility functions will give different results.
## Answer by Mike (score 3)
https://quant.stackexchange.com/a/84044
$0. I don't have an edge, this is a 50/50 betShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.