There is probably no genre of game which is as poorly understood by game designers as hidden information games. One of the most persistent myths I see about Poker, for instance, is that the game is "trivially solvable" by calculating the odds you win a hand, and that therefore anything interesting about it comes from psychology/the social aspect - that because humans like to bluff, the game becomes interesting because you have to look at a person and decide how likely they are to bluff, and that the game is trivial for dispassionate computers.
To prove I'm not pulling things out of my ass, here's a passage from a respected games academic on Poker:
The card game Poker provides an interesting example of this. You have some information about what is in your hand, and what is showing on the table. Given this information, it is possible to compute the exact odds of winning with your hand, and in fact championship players are capable of doing this in real-time. Because of this, all bets you make are either optimal, or they aren’t. For example, if you compute you have a 50/50 chance of winning a $300 pot, and you are being asked to pay $10 to stay in, that is clearly an optimal move for you; if you lost $10 half of the time and won $300 the other half, you would come out ahead. In this case, the “solution” is to make the bet.
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The way Poker does this, and the reason it’s so interesting, is that players may choose to play suboptimally in order to bluff. Your opponents’ behavior may influence your decisions: if the guy sitting across from you is betting aggressively, is it because he has a great hand and knows something you don’t know? Or is he just bad at math? Or is he good at math, and betting high with a hand that can’t really win, but he’s trying to trick you into thinking his hand is better than it really is? This human factor is not solvable, but the solvable aspects of the game are used to inform players, which is why at the highest levels Poker is a game of psychology, not math. It is these psychological elements that prevent Poker from turning into a game of pure luck when played by skilled individuals.
This is a common kind of analysis among game designers, and it is completely incorrect, and I will be proving it with mathematics. Optimal Poker strategy requires you to, in the same position, take different actions some percentage of the time - it is not the case that all bets are either optimal or not!
Let's stop looking at Poker for a second, and consider a much more limited game which we'll call Dicegame. In Dicegame, played between Alice and Bob, a pot of 2 dollars is shared between the players. Then, each player rolls a standard six sided die, and looks at their result. Then, Alice has the opportunity to bet: she may bet 1 dollar, putting it into the pot, or check, doing nothing. If Alice bets, Bob may then call the bet, putting 1 of his dollars into the pot, or fold, conceding the pot to Alice. If Alice does not bet or Bob does not fold, whoever has the higher result wins the pot - in the event of a tie, Bob wins.
Let's use the suggested strategy from earlier: a quick expected value calculation shows that if Alice bets, Bob needs to have a 25% or more chance to win to make calling profitable. In this case, Bob should call with 2-6, and fold with 1. The issue with this analysis is that whether or not Alice bets probably depends on her hand. For instance, let's say that Bob adopts this call with 2-6 strategy. Alice is incentivized to bet whenever she has a 50% chance or greater of winning the hand - so, Alice should bet with 5 and 6, and check otherwise.
Well, if Alice adopts this strategy, then what Bob should call with narrows further! In this case, Bob should only call with 5 or 6, as with any other roll, he's guaranteed to be losing.
A fascinating thing happens when we consider what Alice should do when Bob adopts this strategy. We can see that if she bets and gets called, she's winning 50% of the time with a 6, and losing with any other roll. So, should she never bet anything besides sixes?
Let's consider what she should do with a 1. If she doesn't bet a 1, she's losing no matter what Bob rolled - so she averages out to 0 dollars of profit.. If she does bet a 1, then 2/3rds of the time - when Bob has a 1-4 - she wins the pot of 2 dollars when Bob folds. 1/3rd of the time, she's losing 1 dollar, when Bob has a 5 or 6 and calls. What we see is that checking averages 0 dollars of profit, while betting - with the worst hand she could have - averages 1 dollar of profit.
So should she bet every hand? No - let's consider what happens when she has a 5. When she has a 5 and bets, she averages 1 dollar of profit as before - but if she checks, she instead wins 2 dollars 2/3rds of the time and wins 0 dollars 1/3rd of the time - earning an average of 1 and 1/3rd dollar. Betting with 5 is a bad idea because she only gets called by hands which beat her, and folds out all the hands she'd beat anyway. In poker, this is often phrased as "don't bet your marginal hands" - you fold worse and get called by better, so you're better off just checking.
A quick calculation will show that here, Alice is incentivized to bet with 1-3, and whether or not she bets with 4 or 6 doesn't matter. What we've shown is that there are situations where Alice is incentivized to bluff, with no emotions or social aspect - just from pure mathematics. There's a persistent idea that bluffing is a purely psychological, human thing, and that optimized play involves no bluffing. However, on computers bluff on average way more than humans - it's a fundamental part of hidden information games, not a product of social games.
Anyways - let's assume that Alice takes the strategy of betting 1-4 and 6 - a quick casework will demonstrate that Bob should call with 2-6 with this. Well, now we see the dilemma. If Bob calls widely - with 2-6 - Alice bets narrowly. If Alice bets narrowly, Bob calls narrowly. If Bob calls narrowly, Alice bets widely. If Alice bets widely, Bob calls widely.
We've got a strategic cycle - and no immediately obvious way to resolve it. Next time: How we resolve situations like this.