Maximizing expected value - "triple or nothing" on a fair bet

The "triple-or-nothing paradox" is that a game where I expect to increase my money (on average) at each stage ends up bankrupting me with probability 1 if I play long enough. However, the paradox seems to have an easy resolution, in the sense that a rational actor could decide that a 50% risk of losing your entire stake on each round is simply too high.

A slight modification removes the "easy way out" for this rational actor: I stake an initial wager W to play this game. At each coin flip, I either lose 75% of my current stake, or gain 125% of my current stake, with 50-50 probability. Any individual round of this game is quite attractive to a bettor, for the following reasons:

• at every wager, the amount the bettor stands to win is bigger than the amount they stand to lose

• the bettor never loses their entire stake (if we consider continuous quantities of money) and thus can always make a comeback

• the bettor's expected value of their wager is 1/2 (125% - 75%) = 25% gain

• expected value after n rounds is (1.25)^n of initial stake -- the bettor's expectation increases exponentially

At each round, the bettor has to choose between playing again and cashing out. The arguments above seem to favor playing again for any individual round. Yet over time, the bettor's wealth approaches 0 with probability 1: because it multiplies by 2.25 on wins, but divides by 4 on losses, more than 1 win is required to balance out a loss. Since the flips are fair, this means the bettor's wealth is multiplied on average by (2.25)/4 = 9/16 = 56.25% for every 2 rounds, which translates to a 3/4 = 75% multiplier per round on average. In other words, the bettor's actual wealth (as opposed to their expected wealth) multiplies by about 0.75 on each round, hence approaches 0 for large numbers of rounds.

The paradox is then that during the game, the bettor always has a reason to play the next round in terms of expectation. It is always irrational for them to stop playing after any particular round. Thus, a rational actor will decide to continue playing forever, whereupon their initial stake dwindles to zero with probability 1. How is it that making the most rational decision at each stage (whether one is currently winning or losing) leads to going broke?

Edit: Some commenters have attempted to resolve the paradox by appealing to an individual’s utility differing from the strict monetary payoffs. However, this attempted resolution utterly fails: if we know an individual’s utilities, we can always set up a version of this paradox that applies to them specifically. They “stake” something worth a certain amount of utility to them, and at each stage of the game, either lose 75% of the utility value of their current stake, or gain 125% of the utility value of their current stake. Once the game has been adjusted to their utility preferences, they face the same paradox as before. The presentation in terms of monetary value is for expositional purposes only and in no way fundamental to the paradox itself.

• Comments are not for extended discussion; this conversation has been moved to chat. Oct 8, 2020 at 9:18

The important feature of an example like this is that we are being asked to find the optimal strategy for playing a multi-round game. The fundamental reason that maximizing expected value at each round does not deliver optimal value over multiple rounds is because the process in this example is not ergodic. Ergodicity refers to a property of a system whereby for a given property of interest, its time average is guaranteed to be equal to the ensemble average. A system is non-ergodic when these differ. When a player makes a series of bets and has to bet all their stack at each bet, there is an asymmetry between winning and losing. A loss of one half of the player's stack will require a double to return to the previous level.

If the size of a bet is a controlled variable, optimizing the outcome requires the size of the bet to be adjusted to maximize the expected geometric growth rate, not simply the expected value at each individual bet. If the bet size is fixed, as in your example, and the bet size is above the optimal level of percentage of the stack, then the expected stack size will converge to zero. It sounds odd, but intuitively it is easy to see that it only takes a small run of losses to dig the player into a deep hole from which they are unlikely to escape.

The optimal bet size in such a situation is given by the Kelly Criterion, which maximizes the expected value of the logarithm of the stack. In your example, with constant probabilities at each bet, the optimum bet size would be 53% of the stack. In other words, if the game allowed the player to choose how much to bet at each round, this would maximize the stack size over time. Less than this and the player is making less money than they could; more than this and the geometric rate of return will cause their stack to converge to zero. Since your game requires the player to bet their entire stack, it is not rational to play.

Again, it seems odd, but remember that we are talking about the optimal strategy for playing a multi-round game with a fixed amount of money at the start. Imagine if you were allowed to play this game, but the stake had to be everything you own. It would be foolish to play. Alternatively, imagine you were allowed to keep adding to your stack with an extra buy-in every time the stack got low. Then it would be rational to play, as long as there is no risk of you running out of money altogether. In the limiting case where you are allowed to bet the same amount each round and there is no risk of running out of money, ergodicity is restored and it is rational to play.

The Kelly criterion is used in investment portfolio theory for maximizing expected returns from investments over time and estimating optimal position sizes.

• I upvoted, but this doesn't totally answer: the agent is only making a decision about whether to play the next round, or cash out and take their winnings/cut their losses, rather than a decision about the whole game. The issue is that once this rational agent starts playing, at no individual stage is it rational for them to quit, yet continued play is a losing strategy. I'll also note that as in St. Petersburg, you don't have to put all your money in at the outset, but once you place your stake, what you get is what you get. Oct 7, 2020 at 0:40
• The idea that the geometric mean strategy does better than other strategies in the long run is mathematically fallacious, as was shown by Samuelson, see The “Fallacy” of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling"A maximum-geometric-mean strategy does indeed make it “virtually certain” that, in a “long” sequence, one will end with a higher terminal wealth and utility. However, this does not imply the false corollary that the geometric-mean strategy is optimal for any finite number of periods, however long". Oct 7, 2020 at 8:04
• The use of geometric mean in portfolio selection is also dubious, and Sharpe's ratio is the standard choice instead, see Estrada's recent survey. In any case, even the proponents "admit that this criterion was less general than maximizing expected utility and that it was not the only rational criterion", and since geometric maximization is equivalent to maximizing the expected value of logarithmic utility I do not see how it embodies "rationality". There is no "the outcome" that any one strategy optimizes. Oct 7, 2020 at 8:37
• Samuelson's paper shows that there is no single rule that delivers optimal results over a range of all possible utility functions. But for a long run of bets of the kind envisaged in this question, maximizing the geometric mean will almost surely lead to the highest terminal outcome, and this is what most people are likely to care about. A very few people might get seriously rich from playing this game, but the vast majority will lose their money, so why would you want to play? Oct 8, 2020 at 0:32
• More generally, recent studies of rational decision making are showing that non-ergodic models are providing better explanations of behaviour than expected utility models. For example, of the natural bias towards avoiding very bad outcomes, of the tendency to avoid Knightian uncertainty, and of the preference for avoiding volatility. nature.com/articles/s41567-019-0732-0 At bottom, the answer to the question of why we should not play this game when the expectation value is positive is that maximizing expectation value is not the only important criterion for making good decisions. Oct 8, 2020 at 0:35

Unlike what is said in another response, if, very unrealistically for real people, your utility is indeed linear in the final outcome after finitely many rounds (however many -- say T), then always betting is optimal. A quick proof that that works even if you could skip a round:

1. It is clearly suboptimal to never bet across all T rounds; betting once would do better. So there will be a first round with a bet.
2. There's no reason to postpone that first bet; if you were planning to skip some rounds, might as well move the skipped rounds to the end.
3. Once we arrive in the second round, whatever amount of money we now have, the same reasoning applies.

That being said, it is true that with high probability you will lose a large fraction of your money. If you think about the logarithm of your amount of money and how the bets contribute to that, then the outcomes of the bets are independent and identically distributed, and so the law of large numbers applies. However, because your utility function is actually linear in money (not the logarithm), there just isn't a whole lot to lose. If you start at 1, you can only go down to 0. After a lot of bets, chances are you'll end up very close to 0. But at that point, who cares about the possible downside of taking another bet -- there's now even less to lose. And there's still some chance of getting very lucky over all the rounds and winning a relatively huge amount. That chance is what is making this unrealistic ideally risk neutral agent willing to take these bets.

• I really like the comment that a rational agent will only lose their initial stake, at worst, as an argument (or, at least, a rationalization) for their participation. However, the rational actor also has incentives to invest, if not their entire net worth, an amount equal to (entire net worth - epsilon), for some small epsilon. This is so because the bigger their initial stake, the bigger the payoffs in the event of a windfall, and the less "good luck" needed to win big. So our rational agent pursuing their best interests will lose almost all their money if they play. Oct 9, 2020 at 21:41

Money like many things, doesnt have a linear value. Meaning, you might not want to risk whatever % of your money if you barely have enough to feed yourself. But you may do so if you have disposable income. You can't reduce rationality to such numerals. And despite your edit, no utility benefits linearly with its metric: You could replace money with life expectancy, daily sex, food intake, whatever the utility is, its value doesnt follow its quantity linearly.

Let's make the paradox even simpler, each round you have 99% chance to double your money and 1% chance to loose all. You know you're gonna loose at some point, you just don't know when. Then the game is about weighting if you can afford to loose your initial wager vs what a win will afford you. With that in mind you can find the spot where the game brings you the best value and when you should get out.

With your modification of the game (+125% vs -75%), it's a bit different but not that much. If you play an infinite number of times, your wager may globally tend towards 0 but chaotically enough so you may get out in a beneficial manner (at the spot defined above).

We can make it even simpler then, if right now I ask you to play your wallet's content on a single round of any such game ..will you play it? Can you afford to loose your wallet's amount ?

In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more certain outcome

the wiki on risk aversion goes in great details about these matters

• "Despite your edit, no utility benefits linearly with its metric: You could replace money with life expectancy, daily sex, food intake, whatever the utility is, its value doesnt follow its quantity linearly." That's not the point, because you can just set up the payoffs so that your utility doubles with each win, whatever the specific payoff is. Apr 13, 2021 at 20:07
• See Stanford Encyclopedia of Philosophy on the related St. Petersburg Paradox, where objections to doubling payoffs also arise: "The payoffs need not be expressible in terms of a fixed finite number of commodities, or in terms of commodities at all […] the lottery ticket […] might be some kind of open-ended activity -- one that could lead to sensations that he has not heretofore experienced. Examples might be religious, aesthetic, or emotional experiences, like entering a monastery, climbing a mountain, or engaging in research with possibly spectacular results. (Aumann 1977: 444)" Apr 13, 2021 at 20:11
• "A possible example of the type of experience that Aumann has in mind could be the number of days spent in Heaven. It is not clear why time spent in Heaven must have diminishing marginal utility." plato.stanford.edu/entries/paradox-stpetersburg/#HistStPetePara Apr 13, 2021 at 20:11
• @RiversMcForge have you noticed how difficult it is for you to come up with examples of utility you could double or expand indefinitely? A hammer is useful, 20 hammers not so much. You now come up with a magic hammer that can do the job N times as fast, no matter how fast you make N grow in order to keep increasing utility, at some point, most people wont care. Same goes for days in Heaven (sigh). Apr 14, 2021 at 3:32
• I don’t accept those as examples of diminishing marginal utility, and I should point out that “diminishing marginal utility” is different from “bounded total utility”. Even if the nth day in Heaven only gave me 1/n as much utility as the first, I would get utility proportional to log(n) for all n days, which is unbounded for n large. Apr 15, 2021 at 23:31