Consider a competitive game in which the players each start with 100 points, and repeatedly engage in a positive expected value gamble, with the goal of being the first player to attain 100000 points. Perhaps each round of gambling consists of the player rolling dice, and winning or losing some points depending on their sum. The problem at hand is to prescribe how many points the player should wager each round, and I will make the simplifying assumption that this does not depend on how many rounds have passed or the performance of other players, so that the player's strategy is wager size as a function only of their point total.

The most reasonable approach seems to be imposing a heuristic constraint on the player's risk of ruin, and maximizing wager size (thus minimizing the average number of rounds required to reach 100000 points) relative to that constraint. Let's say the player desires to have no greater than a 2% probability of going bankrupt before reaching the 100000 point goal. Now we can simulate the complete game a large number of times to discover a strategy that fits the bill.

Naturally, risk of ruin will start at 2%, but decline as the player's point total increases. However, when the game is played in practice and the player has accrued, i.e., 1000 points, I see no fundamental reason to suppose her risk preference would have changed from the initial 2% (apart from assessing her position relative to other players, which is beyond the scope of this question). But then she will be rationally compelled to abandon her strategy, and adopt a riskier strategy that has a 2% risk of ruin moving forward from her current, 1000 point game state. Allowing risk of ruin to decline too much as the game progresses would be an unnecessary speed concession.

The problem is that abandoning strategies as the game progresses to retarget 2% risk of ruin means there was actually greater than 2% risk of ruin when the game started. This shouldn't dissuade the player from retargeting (the past is in the past) but foreseeing this from the start we know that the initial strategy is too risky, and therefore not viable. It seems what's needed is a strategy where the risk of ruin at every point total is more or less constant (maybe lower at the very end of the game due to the discreteness of point totals), but such a strategy seems impossible to construct, because attaining points will monotonically decrease risk of ruin in any possible strategy (again, maybe ignoring edge cases).

Is is impossible to construct a strategy for this game without at any point exceeding a 2% risk of ruin (which could more generally be any arbitrary constant)? If not, how can a player whose underlying risk preferences won't change much as the game progresses play this game rationally?

Why should we accept any RoR at all? It seems to me that the optimal wager at no point would ever be allin. You'd always bet some fraction of your roll. With no goal I'd guess you'd just follow the Kelly Criterion and even with a goal you'd probably stick close to that until near the goal.

By risk of ruin (indexed by a particular point total), I mean the probability that a player playing the game alone and following their strategy would go bankrupt before reaching the 100000 point goal, not the probability that they would go bankrupt on the very next wager. I agree that all in would never be prescribed by such a strategy.

If there was no goal, risk of ruin would be 100% as per gambler's ruin. There IS a goal here though (100000 points), so the answer is not so trivial.

I'm afraid the Kelly Criterion doesn't strike me as viable. Kelly prescribes a strategy in which wager size is a linear function (constant percentage) of the player's point total, which already feels suspiciously inadequate to me (wouldn't we expect adding quadratic, cubic, etc., terms to the linear term to improve the tradeoffs between risk of ruin and point accrual speed?). However, the bottom line is that Kelly, like any other strategy I can imagine, allows risk of ruin to decrease substantially as point total increases, so a rational player with roughly constant risk preferences will abandon it before reaching the goal.

Explain to me how you can go bankrupt without placing an allin wager at any point.

If you're allowed arbitrarily precise bet sizes then there's no reason to ever go bankrupt.

You said the game was positive expectation right? Gambler's ruin doesn't apply. On a +EV game RoR is <100% even if you are forced into fixed bet sizing.

I don't think there's really a paradox here. When you improve your position your risk must go down. If you change your strategy to try to maintain a constant risk then your previous risk calculations were incorrect. Any risk calculation should take into account future changes in strategy.

Well the idea of a point is that it is an indivisible unit, but I suppose I should have stated so explicitly. There's no weird fractional rounding to consider with payouts either; everything is an integer multiple of the wager size. I wasn't too worried about rigorously assessing the extreme edge cases, but if the player gets to 1 point, then they would indeed have to go all in to continue.

I have, it seems, misunderstood Gamber's Ruin. I'll have to look into it more at some point, because the idea that risk or ruin is 100% if a gambler never stops gambling still seems intuitive to me.

But as for the paradox, I don't see how we could take into account and mitigate such further changes in strategy during initial strategy creation. Just as a quick example, suppose the purple line is the player's risk preference and the black curve is the game's risk of ruin.



If we shift the black curve up, then it is no longer a viable strategy, because it exceeds the player's risk tolerance at the starting point total of 100. But it cannot remain as is or be shifted down because we can foresee that this would cause the player to shift the curve up in the future anyway (by abandoning the initial strategy and following a riskier strategy). We might be able to adjust the convexity of the curve, but even if becomes linear with negative slope, the problem persists. It needs to be pretty much linear with zero slope to resolve the paradox, but this is impossible, because as you say, when you improve your position your risk must go down.

The idea of gambler's ruin is about playing games of chance with the odds against you, like craps, roulette, and slots in that every player will eventually go broke if they play long enough. It is not true for breakeven or better EV games. In an infinite game of coin flips, the winners and losers approach an exactly normal distribution curve.

Infinite series of ever-decreasing probabilities can easily converge to less than 1. Casinos aren't destined to get bankrupted by players.

I imagine it's ~possible with variable bet-sizing, but designing this without brute force would be difficult. Interesting problem.

Your strategy wouldn't change, only your bet sizes. Your strategy is a set of bet sizes mapped to each possible point total. Your RoR calculation would need to account for the entire tree of possible future states, but I don't see a paradox here.

I think what you're getting at is the correct idea that the player abandoning Strategy A and instead following Strategy B after accumulating some number of points is functionally equivalent to their having constructed a strategy from the start whose actions are identical to A when they would choose to follow A, and B when they would choose to follow B. So the question becomes "What strategy can the player construct from the start which they will not later abandon?"

The paradox is that if at any point total (maybe sans edge cases) their risk of ruin exceeds 2%, the strategy is not valid in that sense because they will violate it and take less risk, but if at any point (maybe sans edge cases) their risk of ruin is not as close as possible to 2% without exceeding it, the strategy is also not valid because they will violate it and take more risk.

The only way I can see to get around this is for the risk of ruin to always match the player's risk preferences, but as browni pointed out, when you improve your position your risk of ruin must go down, while our player's risk preferences are constant.

I change my mind about a constant RoR being possible. But I wouldn't call it a paradox − the idea of it is an outright contradiction. The calculation for r(100) must factor in the future bet-size increases, and so r(n>100) must be less than r(100) because r(100) is a function of every r(n>100).

Are they really, though? If you want a 2% RoR at 500 points then you didn't really want a 2% RoR at 100 points, or you did but now you changed your mind and are seeking more overall risk than originally.

The constant risk preference piece requires a few qualifiers to be precise, but I've been taking it as more or less an axiom. For example, it depends crucially on the player's state relative to other players (being ahead of others warrants less risky behavior, being behind others warrants more risky behavior), but this is too complex a problem for me to solve, so I've stipulated that I am only looking at the problem where such variables are held fixed. My overall aim, therefore, is not a complete solution to the game, but a strategy to use when none of the variables I've excluded are subjectively interpreted to be overriding factors.

Another qualifier is that I don't care too much about my risk preference statements holding in edge cases, such as for point totals very near the end of the game. Discreteness will undoubtedly make some weird things happen there.

Finally, risk preference might actually be a range (i.e., 1.7%-2%) rather than a single constant, so maybe the strategy's risk of ruin need not be exactly constant. Still, it seems to me that the risk of ruin in any strategy will vary wildly across the spectrum of point totals, not just by negligibly small amounts.

The player changing her mind about risk is where the problem comes in, but it is a warranted change of mind. It makes little rational sense for her to care that her present decisions affect her past probabilities of ruin, as the past is already fixed in the past.

The past is already past, but also your future bet sizes will impact the present RoR. If you care about your present RoR then you must care about the effect your future decisions have on it. You can't even avoid thinking about the future because any RoR calculation you make will include assumptions about your future decisions (and therefore your future preferences too).

I think this makes more sense, but I agree that our control of the tightness would be limited. You can make 2% the max, or perhaps engineer some kind of average RoR of 2%, but you can't make the range arbitrarily tight.

So I think at best this idea might abstract but not resolve the conceptual paradox/contradiction, but would something in the neighborhood of a power series serve as another way of approaching the construction of a strategy?

As I mentioned above, Kelly Criterion is a linear function of a player's point total, but yields a very non-constant risk of ruin. Suppose through simulation we see if adding a constant number of points to each wager improves the strategy (now an affine function of the player's point total) in terms of its risk of ruin constancy. Then add to the affine function a cubic term, etc., simulating the resulting risk of ruin profile at each stage. This would be inspired by building a function from its Maclaurin Series. Is this ever a good way to build a strategy, or do area of convergence concerns ruin it? If so, would a similar approach using, i.e., a Fourier Sine Series work better, since such series approximate functions over a region?

I still think rounding Kelly to an integer bet size would be optimal or extremely close. In the OP you claim the most reasonable approach is to "impose a heuristic constraint on the player's risk of ruin, and maximizing wager size," but I don't understand how you arrived to that conclusion.

This could probably be settled with computer analysis. You seem to want to ignore the race aspect and simply find the strategy which minimizes the expected number of wagers to reach the goal, correct? Or is the point not to solve the game but demonstrate this paradoxical concept you think you've found?

I forgot to account for RoR. A strategy would have to be evaluated by RoR and the expected number of wagers given you didn't hit ruin, but then there's really no way to say something like 5% RoR with 1000 expected wagers is better or worse than 2% RoR with 2000 expected wagers, right?

I don't understand why we started with the race idea and immediately discarded the aspects that make it a race.

Yeah I mean the premise of the question is flawed: A rational player would care about maximizing their chance of victory, not targeting a specific ROR. The latter can only be rational if you're just playing on your own and not involved in a race (and if you have strange preferences).

If we're trying to win the game, I too think Kelly is probably the answer. And its initial RoR would be about 1% since that's roughly the chance of being reduced to 1 point when starting with 100.

There is no way to state that one or the other is absolutely better. In practice, it will depend on the player's point total relative to others'. The reason I exclude this dimension from my analysis is that solving the full problem strikes me as too hard. I'd need to introduce some pretty heavy game theory, such as the impact of other players copying the player's wager sizes, how rationally other players will act, etc.

Most of this is IMO easier to execute subjectively, so I am using the "minimize expected number of wagers with a risk of ruin constraint" heuristic as a simplification. The prescription then would be to estimate subjectively the probability of being outpaced as the game progresses, and switch to a corresponding strategy with slightly lower risk of ruin. This would also mess with the initial probabilities in a way comparable to what I've described in this thread, but I don't expect optimality in the subjective aspects of game play.

What relevance does all my talk about constant or near constant risk of ruin preferences have then? It would apply, for example, if the player pulls far ahead of the competition. The player would want a low risk of ruin, more or less independent of absolute point total, but the pace still matters so it shouldn't be too low.

What do you do in the event Kelly yields a .05% risk of ruin from your current position forward? Keep chugging along at a slower pace than the competition?

It's not a slower pace, it's the maximum rate of growth. Betting bigger will slow you down on average.

But you're right that we'd also need to consider the relative scores, as well as our distance to the target. If we have 990k then there's no sense in wagering 100k when all we need is another 10k. And intuition says we'd wanna be more aggressive after falling behind and perhaps more conservative when ahead, though I'm not certain.

By "on average," are you re-counting the outcomes in which ruin occurs? If not, then I think you are implying that the average number of rounds required to reach the 100000 goal given that you eventually reach the goal (rather than ruin) does not monotonically decrease as the bet size at every point in the strategy is increased, but rather has a minimum which Kelly approximately picks out?

I mean the geometric average. If I bet kelly and you bet 1.5x kelly, my plot of score vs time will be a steeper exponential curve than yours. (But if it's a close race as the players approach the finish line, then it can make sense to bet >kelly.)

As for the arithmetic avg time, that's infinity due to the nonzero possibility of ruin. If it's a given that we'll reach the goal, the time is minimized by betting all-in each round.

Okay, this is helpful in terms of further exploration. I need to do more simulation, as I had assumed from the data I observed that the larger the bets the quicker the player reaches their target, as long they don't get ruined first.

The green and red statements strike me as mutually contradictory. In particular, the green statement is much less intuitive to me than the red one. I tried producing both effects in C# code, but was only able to confirm my intuition that larger wagers produce steeper score VS time plots.

First to confirm the red statement, I ran a simulation of a game 1 million times for each integer percentage wager size as a linear function of the player's score. The game consists of rolling three dice per round, where the player loses their wager if the result is 10 or less, receives their wager as a net gain if the result is 11 through 14, nets double their wager for 15, nets triple their wager for 16, and nets quadruple their wager for 17. I ran a version of this game with a large jackpot in which the player nets 99 times their wager for 18, and a no jackpot version in which the player nets only 5 times their wager for 18. The x-axes represent percentage of score wagered each round, and the y-axes represent expected number of round required to reach the goal. The results were as expected (here is the code if interested: https://pastebin.com/r1PBjMtk)...

Jackpot



No Jackpot



Now for the green statement, I ran a simulation of the same game 1 million times, but instead of iterating over different wager sizes, I created a separate score VS time (number of rounds passed) plot for each wager size. Here I computed the round-wise average scores for the first 100 rounds, so that the x-axes represent number of rounds passed, and the y-axes represent expected score. I removed the victory check from this version so that the steepness of the pure exponential growth can be observed. The results seem to disconfirm the green statement (here is the code if interested: https://pastebin.com/etvvcv57)...

Jackpot











No Jackpot

I won't post these images but they show the same thing. I also omitted a lot of the images I made in order to prevent clutter, but what can be seen in all cases is that expected score VS number of rounds passed always gets steeper as wager size is increased. I tested this up to a wager size of 50% of the player's point total, which produces ruin far too often (about 95% of the time) to be viable, and never encountered a maximum steepness that could correspond to the wager size Kelly would pick out.

Is this experiment designed wrong? How can I understand the apparent contradiction between the green and red statements, and the counter-intuitive (and apparently empirically disconfirmed) nature of the green statement in its own right?

The red statement was silly because it's trivial. If a genie grants that you'll never be ruined no matter your bet-sizing, then of course all-in is optimal because you'll never lose a bet.

The green statement accounts for the lack of a genie. Losing wagers will slow your growth down if not ruin you altogether.

If you're talking about arithmetic averages then that's what's wrong with the experiment. The arithmetic average score is maximized by maximizing EV, ie by going all-in. However, the green text is a statement about the geometric average.

With a jackpot, the kelly bet is 12.29% and achieves a geometric avg growth factor of 1.0182, ie a plot looking like 1.0182^x. No higher rate can be achieved with a different bet size.

For the experiment, you can divide score(time)/score(time-1) each sim, multiply all the fractions together and take the Nth root where N is the # of sims.

Shouldn't the 7 be a 6? The 7 increases the Kelly bet from 11-12% to 41-42%.

There shouldn't be a 95% ROR when only betting 1.22x kelly and only playing 100 rolls. More like 23%. Did you use Next(1,6) for this set of sims? That would explain it.

The green text is just a statement about the steepness of the graph; I think the arithmetic and geometric averages you're referring to are actually derivable from the same graph, meaning that the green statement is indifferent between the two? When I was talking about round-wise average scores, I meant that I played the game many times and computed the expected score after round 1, expected score after round 2, etc., which was how I pieced together the graph. It was an average over instances of playing the game used to remove game-to-game volatility, not a temporal average over rounds.

Nope, Next(1, 7) is how you roll a six-sided die in C#. It uses an unsightly format where the first parameter is inclusive and the second parameter is exclusive, so Next(1, 7) can return 1, 2, 3, 4, 5, or 6.

Definitely not. A plot of the EV is not the same as a plot of the geometric mean value. When you plot EV vs time, obviously it's steepest when the player maximizes EV.

It's my fault for talking about steepness to begin with because that doesn't have a single meaning except when looking at a single experiment.

That's what I figured. You can do an equivalent with geometric averages by comparing each round's score to the initial score. The geometric mean of all the Round 10 ROI's would tell you, "On average, after Round 10 my capital will have increased by x%." For accuracy, you might need to start with more capital than 100 so that the ruinous scores (capital<10 as you defined them) are approximately zero compared to the initial capital.

Alternatively, you can calculate the median score of each round, which is also maximized by kelly. Just the plain old median, not some geometric version.