Expected Value
A basic idea in game theory is the concept of expected value. Intuitively, you can consider the various outcomes that could occur, then sum the value that you gain from each outcome multiplied by the probability of that outcome.

For example, if the game is flipping a coin and you get $50 by winning the flip, and lose $20 by losing the flip, for a fair coin, you could say the value of the game is 0.5*50 - 0.5*20 = 25 - 10 = 15. Given the opportunity to play the game many times, you should do it, because on average, you should expect to be getting about $15 profit each time.

How You Play Go
When I play go, I sometimes think of the game in this manner. For example, I am in a situation, and I know a particular move will not work. But if my opponent screwed up and the move did work, maybe I get a lot more points than I would lose by playing a move I know is correct.

Still, given the idea of expected value, provided I can assess the probability of my opponents playing a certain way with some degree of accuracy, I should on average, gain more profit this way, right?

So what do you think? Is it more profitable to play in a way that will give you a greater expected value, even if the move doesn't work, or is it more profitable to play in a way that you know is "correct" all of the time, even if you may miss out on potential profit that may be likely if your opponent doesn't know how to respond correctly?

This probably applies more to playing against people that are weaker than you, since that would be a scenario in which you'd more likely be able to correctly assess whether they know the "correct" answer to a given position.

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be immersed

In game theory, expected value is distinct from expected utility. Expected value takes into account the average of N different outcomes weighted by the probability of each outcome and by the raw numerical value associated with each outcome. But expected value does not actually describe how people make decisions. For that we need additional information; how much the raw numerical value means to the person evaluating it.

For example, in a bet with possible dollar outcomes of 100,000 and 1,000,000, the utility associated with the outcomes might be 1 and 2 (if you need ten times as much money to be twice as happy).

In go, the numerical values of possible outcomes might be -2, +5, +50; but these would be weighted at 0, 1, 1 when calculating utility, assuming all wins are 1 and all losses are zero.

Sure, utility can vary depending on what something is worth to you.



I'm not sure if I take that approach. Yes, go is a zero sum game, so a win is all that matters. But evaluating points is a common technique for estimating the best ways to play - and it doesn't have to be "points of territory". You can express this estimation of how well you are doing based on estimates of influence on the board, etc.

That is, typically, people try to play moves that give them the greatest increase in the chances of winning, which can be estimated by the value they think they'd gain from doing a particular action.

If there are two places to move on the board, one that will gain me 50 points, and one that will gain me 2 points, a heuristic approach may be to select the area that gains me 50 points.

Unless you are able to read to the end of the game, you cannot know whether a particular play guarantees a win or loss.

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be immersed

But the point is that if you are considering multiple choices, each of which could lead to more than one outcome, maximizing the expected value of the game and maximizing the expected utility give different results. (If your choices aren't risky, there's no need for EV or EU.)

For example, let's say one move has 1/3 chance each of outcomes -8, +3, and +4, while the other move has 1/3 chance each of -2, -1, and +30. EV says that the expected score is -1/3 with the first option and +9 with the second, but for me, at least, the expected utility of the second option is much lower.

If the question is "should I play hamete that I think my opponent will fall for?"

I will give the standard answer.

Playing good hamete- good moves that are really perilous to answer when you don't know them-is totally fine.

Playing a move you know is bad in the hope that your opponent doesn't read or know the refutation is bad.

Playing a correct move versus ignoring your own reading and gambling your opponent errs...

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Revisiting Go - Study Journal
My Programming Blog - About the evolution of my go bot.

Good example. For a single move, there's a 2/3 chance of getting a positive result with the first option, and only a 1/3 chance with the second.

But let's say that there are 100 such choices throughout the game. With the second option, you get a gain of 30 points 1/3 of the time. Doesn't seem so bad, does it?

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be immersed

Sounds like a standard answer, but why is this the case? Why is it bad? That is to say, if you feel that you increase your chances of winning the game by doing so, why not go for it?

An example that comes to mind is computer bots for poker. It's not uncommon for them to try to learn and exploit human irrationality to get a greater profit when gambling. They don't play the "optimal" choices given their cards based on the straight up probability of getting certain hands. Rather, they try to maximize gains by learning human playing styles.

Is this type of strategy bad?

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be immersed

If a trick move increases the chance of winning a game, and a good strategy is one that has the highest chance to win the game, then playing the trick move is a good strategy. If it is good to play a good strategy, then playing the trick move is not bad. So the reason some would say trying to trick your opponent must have little to do with if it makes winning the game easier or not. But I think it is easy to think of strategies that would clearly be wrong even if they increases probability of winning, so the impact on probability of winning a game can not be the only relevant factor when deciding a move.

I was treating those as outcomes of the game, not as values of a single move. If you don't have some idea whether you are ahead or behind, then you're right - you just have to play moves that seem to be big and hope it all works out in the end. But IMHO, it's easier to count the board than to assign probabilities to possible values of moves.

This kind of play only works when the opponent knows less than you do.
Sure you will win more games against weaker opponents, so if you usually play weak opponents for big sums of money, then I'd say go for it...

Playing moves you know to be substandard is however not a good way to improve.
Improvement is the goal, not winning.

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"The connection between the language in which we program and the problems and solutions we can imagine is very close" -- Bjarne Stoustrup

This whole issue ls more complex than indicated so far. Whether a particular risky move is a good bet depends upon the state of the game.

Let us suppose that I have the option to play a tricky move, such that I will lose 5 points if my opponent sees the proper reply, but that will net me 50 points if he misses it. Let us further suppose that I expect my opponent to see the correct reply 99% of the time. The net value of such a move approaches -5.

To play such a move in an even fuseki would be stupid. But if I were down 20 points in yose, if is reasonable.

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If you have no way of evaluating the value of playing a move, how is it possible to "count the board"?

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be immersed

Good point. I wonder if value can be expressed some way as a function of points, current score, and maybe potential left in the game...?

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be immersed

Yes, this is another good example. The EV of such a move is -24.55, but the EU (if we assign 0 to a loss and 1 to a victory) is 0.01, which is slightly better than a move that will gain me 19 points for sure (EV -1, EU 0.00).

(Now, you might assign a certain value to dignity, as well. I probably wouldn't play a move which would insult my opponent's intelligence that much. And likewise, it would be rare to be 100% sure that the alternate move leads to a one point loss. But as an example of how EV and EU can differ, it works.)

I didn't say I had no way of evaluating the value of a move - that's not easy, but as you say it's no harder than counting the board. It's assigning probabilities to different outcomes when you're not sure that is hard.

For example, let's say you have a very standard choice - reduce or invade. It's easy enough to estimate the score if you reduce, or if you invade and live, or if you invade and die. What's hard is to figure out the probability of invade and die versus invade and live.

I'm referring to a situation where you know the correct result, but there's a good chance that your opponent does not.

So in your example, you know the "value" after you invade - say you get +50 points if you invade and your opponent screws up and lets you live. And you also know that if your opponent plays correctly, you get -20 points.

Given your opponent's rank, you may be able to estimate, "at X-kyu, there's probably like an 80% chance that he doesn't know how to kill me".

It's a rough estimate, but you could perhaps say that it's worth 0.8*50 - 0.2*20... However, as Joaz pointed out, issues such as how late in the game it is could affect the value of such an estimation.

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be immersed

I should also say that I may be pressing this idea a bit too much, but that's because I actually feel this way during games. It's not as precise as an expected value calculation, but I often am faced with a situation where I feel, "I know is isn't the correct play, but if I play it there's a good chance my chances of winning will improve".

So sometimes I play the incorrect move.

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be immersed

It all depends on what your goal is in playing. If your goal is just winning then there may be times when you should play a move you know is bad. If your goal is to play the best possible move in the abstract sense then obviously you should never deliberately play a bad move.

Right. I am saying that your count of the board is probably pretty good, whereas "Oh he's 3k, 80% of 3k don't know this trick" is a wild-ass guess. If you are going to calculate an expected value on the basis of a wild-ass guess, you should be willing to go the extra step and make an educated guess about the count.

sounds like a ko threat to me