Without calculations I'd go with betting twice in two lotteries because there's a chance of winning 20 GBP if you win in both but only a maximum of winning 10 GBP if you bet in only one lottery.

(As a useful oversight, the expected return is equal to the bet, so as an additional question, is there a reason to play at all?) — Michael

In commercial lotteries the expected return is much less than the bet, so if your utility function is just the expected return, then most lotteries are losers. And yet people buy lottery tickets. Which means that utility for those people is more than just the expected return.

So the answer to your question:

So, given that the expected return is the same, is there a reason to prefer one way over the other? — Michael

Depends.

Allow me to tweak and probe the hypothetical slightly:

Assuming that we *must* play, what are the consequences of going bust?

In other words, what is the utility difference between my starting chip count and zero, and what is the utility difference between my starting chip count and double my current chip count?

In situations where losing (ending up with 0$ or a very low amount) is disproportionately bad, I would play both games to reduce the chance of going bust. If given the option, I would place as many single bets on as many games as possible to ensure that I would win at least one of them.

Another situation might be that I need to double my current chip count in order to afford some necessary utility, where it's pretty much all or nothing. In this case, betting everything on a single shot would actually be superior because it is a more likely path to a large chip-count increase.

There's a real world example that fits pretty well here: Chinese businessmen looking to convert Yuan to USD while avoiding government regulations purchase a $100,000,000 vacation package at casino resorts in Macau. The resort package comes with 100 million of complimentary casino chips, but to make them redeemable (in USD), they must be risked at least once at the casino. They will usually sit at an ultra-high stakes black-jack table (or something simple like roulette) and will bet a small portion across numerous rounds until all their chips have been risked. It's pretty easy to understand why they don't risk everything on a single hand; even though they could double their money, the risk is too great, and they aren't interested in increasing their wealth (they just want to move it around). Making a single bet would defeat the entire charade.

I made a thread a few years back about more or less the same strategic ambiguity, although I complicated it with individual vs collective utility considerations.

So, given that the expected return is the same, is there a reason to prefer one way over the other? You're more likely to win if you place two bets on one game but you have the chance to win a bigger prize if you place one bet on each game. — Michael

↪Michael Without calculations I'd go with betting twice in two lotteries because there's a chance of winning 20 GBP if you win in both but only a maximum of winning 10 GBP if you bet in only one lottery. — Benkei

In commercial lotteries the expected return is much less than the bet, so if your utility function is just the expected return, then most lotteries are losers. And yet people buy lottery tickets. Which means that utility for those people is more than just the expected return. — SophistiCat

People's utility functions with the lottery can't resemble expected gain, then. Assuming it's a monetary return required, the cost of investing in any single bet is negligible but the possible return is comparatively huge. If you're spending $1 per week on the lottery and can continue that indefinitely, and it really is a negligible cost, then effectively you're paying nothing to be exposed to the small chance of a relatively large payoff.

If you place both bets on one game then your odds of winning are 0.2. If you place one bet on each game then your odds of winning are 0.19. — Michael

I was never any good at math. How does this work out exactly? In my mind, by playing the two games instead of one, someone or something just seems to magically lower the chance by .01 for no rhyme or reason.

Irrespective of one or more unusual and unlikely scenarios described in this thread, the average bloke can afford to toss away say $1 (or so) with the 1/10 chance of netting an additional $9. Why not right? That's how it gets enough to give away such large amounts in the first place. Of course, most lotteries offer far larger prizes.. granted at an extremely smaller chance of winning. And still, the tax or some profits (up to 40% I hear) go toward education and other universally beneficial causes. A win-win either way, no?

I was never any good at math. How does this work out exactly? In my mind, by playing the two games instead of one, someone or something just seems to magically lower the chance by .01 for no rhyme or reason. — Outlander

If you buy one ticket in each game then for each game you have a 0.1 chance of winning, which means that in each game you have a 0.9 chance of losing.

The chance of losing both is then 0.9 * 0.9 = 0.81, which means that the chance of not losing both (i.e. winning at least one) is 1 - 0.81 = 0.19.

In situations where losing (ending up with 0$ or a very low amount) is disproportionately bad, I would play both games to reduce the chance of going bust. If given the option, I would place as many single bets on as many games as possible to ensure that I would win at least one of them. — VagabondSpectre

If you want to ensure that you win at least one then you want to place all your bets on a single game as there's a better chance of winning.

If you want to ensure that you win at least one then you want to place all your bets on a single game as there's a better chance of winning. — Michael

If I could cover extra bases in a single game, then yes. (are we buying tickets to be drawn or selecting numbers on a wheel, or are we buying scratch tickets?).

To answer one of the main questions from your OP, without additional context there is no way to say which one is better. Given that the theoretical risk/reward ratios are functionally the same, it seems that we can only appeal to the needs/desires/circumstances of the gambler to rationally discriminate between the two options in terms of "value"...

is there a reason to play at all? — Michael

For instance, the sheer thrill of risk and the feeling of being a lucky winner (or even just the flashing lights of the lottery machine that have become associated with those feelings). I would guess these are the most popular causes of gambling addiction.

People's utility functions with the lottery can't resemble expected gain, then. Assuming it's a monetary return required, the cost of investing in any single bet is negligible but the possible return is comparatively huge. If you're spending $1 per week on the lottery and can continue that indefinitely, and it really is a negligible cost, then effectively you're paying nothing to be exposed to the small chance of a relatively large payoff. — fdrake

I wonder if this is also the case of the two envelopes problem we discussed a while back. After opening your envelope and finding £10, the expected return when you switch is the same as the expected return when you don't switch, although one's personal utility function may be that you're willing to switch for the possibility of winning £20.

Seems the same as this case of the lottery, where the expected return for placing two bets on one game is the same as placing one bet on each game (and is equal to the bet), but where there's the possibility of winning the greatest amount if you place one bet on each game.

In both cases, there's no purely rational decision.

In both cases, there's no purely rational decision. — Michael

Well, how would you define a rational decision? Any reasoned decision is anchored in values, and values as such are not rational (I don't think). I would say that as long as a decision is explicitly aimed at achieving that which you value, then it is a rational decision. If you like to gamble for the highest potential reward, then it is rational for you to play two different lotteries. If you are risk-averse, then it is rational to play one.

Well, how would you define a rational decision? — SophistiCat

I suppose one where you just look at probabilities and payouts to determine expected returns. If the expected returns are the same then there's no rational preference to play one way or the other, and if the expected return is the same as the bet then there's no rational preference to play or to not play.

Any preference to play, or to play one way over the other, will likely involve something like "hope" for a big win, which isn't a rational reason.

I suppose one where you just look at probabilities and payouts to determine expected returns. If the expected returns are the same then there's no rational preference to play one way or the other, and if the expected return is the same as the bet then there's no rational preference to play or to not play. — Michael

Any preference to play, or to play one way over the other, will likely involve something like "hope" for a big win, which isn't a rational reason. — Michael

I think whether it's rational or not depends less on what someone does, but why someone does it. If someone's utility function is "negligible cost for exposure to relatively large gain", it becomes a rational decision to play the lottery as it's utility maximising. If being "utility maximising" is the sign of rationality, anyway.