Using "infinity" is a cheat, not a handwave. It's a variant of Pascal's Wager to put the thumb on the result you want.<p>"No better off materially" is also something you cannot handwave. There is a limit. If someone gambled all of their money then they would be worse off materially, and quickly. They are not in equilibrium, which means your hypothesis requires some sort of lower bound.<p>This is a well-studied topic. <a href="https://link.springer.com/article/10.1007/s10899-010-9194-0" rel="nofollow noreferrer">https://link.springer.com/article/10.1007/s10899-010-9194-0</a> concludes:<p>> Empirical knowledge on lottery gambling has increased significantly over the past decade. Recent literature on lottery gambling involving numbers games, lotto, and scratch cards has provided three tentative answers to the question as to why people buy lotteries: some people do not behave in a rational way while gambling on lottery; lottery gambling is for fun; and lotteries are so common they are not viewed as gambling. Lottery gambling theories, classified into one that deals with judgment under uncertainty and another that deals with irrational beliefs, continue to be the theories of choice in lottery gambling research. Theoretical frameworks other than those of cognitive theories, such as social cognitive theory and theory of planned behavior, have been introduced in lottery gambling research. Dimensions of personality have also been found to relate to lottery gambling.<p>Your model does not include the "fun" part.<p>You can see that people in the lowest income bracket are the least likely to gamble, I'm guessing because they will be in a materially worse situation by gambling:<p>> The inverted U-shape distribution of lottery gamblers among five SES quintiles (low, 2nd, 3rd, 4th, and high) showed larger proportions of lottery gamblers in the 2nd and 3rd quintiles (70% each), whereas the low, 4th, and high quintile accounted for 61, 65, and 63% respectively (Welte et al. 2002; Table 2).