3 ms·

It's an interview question, you're free to ask for clarifications. That said, finding the Nash equilibrium is precisely the natural thing to do if you can't ma

It's an interview question, you're free to ask for clarifications. That said, finding the Nash equilibrium is precisely the natural thing to do if you can't make assumptions about your opponent's strategy. It's a worst case scenario analysis. You can't statistically lose out to your opponent if you pick it. This does not mean that you know your opponent strategy. Consider the context. This is algorithmic trading. If your strategy can be gamed, it will be gamed by someone else. Picking anything but the Nash equilibrium means you can be gamed. A good candidate will immediately gravitates towards this solution, there is no ambiguity that this is the expected answer.

foldr 10y ago
In the absence of any information about your opponent, you should assume that they're equally likely to be using any strategy. >You can't statistically lose out to your opponent if you pick it. Right, but you can miss out on an opportunity to fully exploit the fact that your opponent is using a really bad strategy. So if you want to maximize your probability of winning against an opponent with a randomly chosen strategy, it's not the right answer. Nothing in the statement of the question indicates that you are entitled to make any assumptions about your opponent's strategy. And it's not as if determining the best strategy in a context where you lack perfect knowledge of your opponent's strategy is an unknown concept in game theory: https://www.economics.utoronto.ca/workingPapers/UT-ECIPA-ECPAP-95-04.ps https://www.economics.utoronto.ca/workingPapers/UT-ECIPA-ECP... I feel like the way you state the question at present may end up selecting for candidates who think exactly like you do. Why not just add the statement that the players have mutual knowledge of their rationality? That removes the ambiguity, and doesn't make the question fundamentally any easier.
- murbard2 10y ago
  1) There's no reason why you should assume every strategy is equally likely. It's absolutely not the case. When you play poker, do you assume your opponent play their cards randomly just because you don't know their strategy? It's not a realistic prior at all. 2) Playing the Nash equilibrium means that you may not be fully exploiting the opponent, true. It's also possible that you're opponent's strategy is to give you a million dollars if you say "abracadabra". So how about that? Check-mate interviewer. 3) If you want to make the problem harder on yourself, you can be my guest and try and assume a realistic distribution of strategies for your opponent. That distribution will have a peak around the nash equilibrium, so you'll have to end up calculating it.
  - foldr 10y ago
    Your way of looking at it is perfectly sensible. But a sensible person might also say that a uniform prior over strategies is appropriate in the absence of any information about your opponent. Choice of priors is always going to be partly subjective, and reasonable people can make different choices. I would point out that in a real world scenario, a uniform prior is barely any less realistic than the a prior derived from the assumption of mutual knowledge of perfect rationality. Is your poker opponent equally likely to be doing their utmost to lose as to win? No. But on the other hand, they're not likely to be tracking 100% of available information and making the mathematically correct choice all the time. In general, the question of what you should assume about an unknown opponent's strategy is a tricky philosophical question, and reasonable people can gravitate to different points of view on this issue. I think that might be why so few people give the answer you want. I was just suggesting that explicitly mentioning mutual knowledge of perfect rationality might help.
    - murbard2 10y ago
      Do you think it's equally likely that a random opponent will redraw below 0.01 as it is that they will redraw below 0.5 or below phi? Most people do not give the right answer not because of philosophical issues around the Nash equilibrium but because they have a hard time understanding that looking to maximize the expected value is not optimal.
      - foldr 10y ago
        >Do you think it's equally likely that a random opponent will redraw below 0.01 as it is that they will redraw below 0.5 or below phi? If random really means random, then that's presumably true by definition. If we're talking about an opponent randomly selected from a representative group of people, then no, but equally, I wouldn't assume that they're going to pick the golden ratio as their threshold either. The thing is that maximizing the expected value is optimal in the absence of any information about the opponent's strategy. But sure, I can easily believe that many people would get the question wrong anyway.

foldr 10y ago

In the absence of any information about your opponent, you should assume that they're equally likely to be using any strategy. >You can't statistically lose out to your opponent if you pick it. Right, but you can miss out on an opportunity to fully exploit the fact that your opponent is using a really bad strategy. So if you want to maximize your probability of winning against an opponent with a randomly chosen strategy, it's not the right answer. Nothing in the statement of the question indicates that you are entitled to make any assumptions about your opponent's strategy. And it's not as if determining the best strategy in a context where you lack perfect knowledge of your opponent's strategy is an unknown concept in game theory: https://www.economics.utoronto.ca/workingPapers/UT-ECIPA-ECPAP-95-04.ps https://www.economics.utoronto.ca/workingPapers/UT-ECIPA-ECP... I feel like the way you state the question at present may end up selecting for candidates who think exactly like you do. Why not just add the statement that the players have mutual knowledge of their rationality? That removes the ambiguity, and doesn't make the question fundamentally any easier.

murbard2 10y ago
1) There's no reason why you should assume every strategy is equally likely. It's absolutely not the case. When you play poker, do you assume your opponent play their cards randomly just because you don't know their strategy? It's not a realistic prior at all. 2) Playing the Nash equilibrium means that you may not be fully exploiting the opponent, true. It's also possible that you're opponent's strategy is to give you a million dollars if you say "abracadabra". So how about that? Check-mate interviewer. 3) If you want to make the problem harder on yourself, you can be my guest and try and assume a realistic distribution of strategies for your opponent. That distribution will have a peak around the nash equilibrium, so you'll have to end up calculating it.
- foldr 10y ago
  Your way of looking at it is perfectly sensible. But a sensible person might also say that a uniform prior over strategies is appropriate in the absence of any information about your opponent. Choice of priors is always going to be partly subjective, and reasonable people can make different choices. I would point out that in a real world scenario, a uniform prior is barely any less realistic than the a prior derived from the assumption of mutual knowledge of perfect rationality. Is your poker opponent equally likely to be doing their utmost to lose as to win? No. But on the other hand, they're not likely to be tracking 100% of available information and making the mathematically correct choice all the time. In general, the question of what you should assume about an unknown opponent's strategy is a tricky philosophical question, and reasonable people can gravitate to different points of view on this issue. I think that might be why so few people give the answer you want. I was just suggesting that explicitly mentioning mutual knowledge of perfect rationality might help.
  - murbard2 10y ago
    Do you think it's equally likely that a random opponent will redraw below 0.01 as it is that they will redraw below 0.5 or below phi? Most people do not give the right answer not because of philosophical issues around the Nash equilibrium but because they have a hard time understanding that looking to maximize the expected value is not optimal.
    - foldr 10y ago
      >Do you think it's equally likely that a random opponent will redraw below 0.01 as it is that they will redraw below 0.5 or below phi? If random really means random, then that's presumably true by definition. If we're talking about an opponent randomly selected from a representative group of people, then no, but equally, I wouldn't assume that they're going to pick the golden ratio as their threshold either. The thing is that maximizing the expected value is optimal in the absence of any information about the opponent's strategy. But sure, I can easily believe that many people would get the question wrong anyway.