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How to Win Every Game - PAX Prime 2014
How to Win Every Game
PAX Prime 2014
It seems obvious that, when playing a game of skill, one attempts to win. Interestingly, this is often not the case, and even skilled gamers rarely analyze to any real depth the underlying mechanics and strategy of a given game. By deconstructing the games we play, you too can make them far less fun for yourself and beat the everliving hell out of your friends. We'll hit the theory pretty heavily, but also specific examples from games like Stratego, Settlers of Catan, and even Football.
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If someone is generally considered better at combat, most people want to gang up on them from the start.
If someone is on 4 points, EVERYONE gangs up on them to make sure they don't get the 5th point.
If two or more people are on 4 points, the same, but then the two on 4 points try to get cheap knockouts on each other really early on in the round.
If there are only two players left at the end of a round, and one has 4 points and the other less, those already knocked out tell the lower scoring player to take a hit for the "team" and make the 4 point player drop even at the expense of gaining a point. Sometimes a kamikaze attack is safe than juggling at the end.
http://www.sfbrp.com/archives/707
Alternatively, I would argue that by virtue of being a human being rather than a game-theoretic "rational agent" it's possible to genuinely communicate a serious intention to follow through with your threats. At worst, you could use some kind of commitment device.
Here's the thing: within a game, there is nothing you can do outside of the game to establish that reputation. If it makes more sense for you to be bluffing, people will think you're bluffing. The only way to commit is to actually commit, and prove there is no bluff.
Besides, you can even establish a reputation of this sort within the context of a single game. If you take multiple opportunities to make a threat, and on the first time follow through with that threat even if it would be apparently "irrational" to do so, then your opponents should take your subsequent threats at least somewhat seriously.
So you are then manipulating the non-bluff by tricking your opponent into thinking you are acting randomly, when not. No kind of reputation helps in this case, because it falls outside of what a reputation is for.
Really, throwing the steering wheel out the car is the ONLY way for the opponent to be sure you have committed 100%.
Second, I specifically said "In a single game with no iteration, there is no way for an opponent to know your track record." You can't now say "If you take multiple opportunities to make a threat, and on the first time follow through with that threat"... you've simply taken the argument outside the scope of my point, and the point of the argument presented in the video.
There is a big reason why iterated prisoners dilemma is won by different algorithms than single shot games.
Also, Luke is spot on with the difference between a bluffable threat and an irrevocable/credible threat. All threats are potentially bluffs unless you literally already pre-took your action. Most games don't allow this: your move next turn in Risk is not determined mechanically until that turn: anything you say or do before that turn means literally nothing. All that matters is, on your actual turn, the moves you actually make.
Threats can be socially credible in Risk, but should be ignored a percentage of the time like any other strategy would entail: there is no way to escalate to an irrevocable threat.
As for "bluffing" in general, we were talking about a scenario where you want to credibly precommit to taking some kind of action. However, as far as the pure game theory with so-called "rational agents" concerned, the words that come out of your mouth in that situation are arbitrary and meaningless. If there was no point in your saying whatever you said in the first place, why does it make more sense for them to be a bluff?
What kind of bluffing are you talking about?
If someone makes a blatant, visible display of apparent randomness then that's an important sign. If they just wanted to be unpredictable all they had to do was pick randomly; there is no need for them to let me see it. As such, there has to be an additional reason why they wanted me to see their display of apparent randomness, and I'm obviously going to spend at least some time thinking about what that reason is. Nope, reputation still matters here. If you don't have a reputation for making genuine visible displays of randomness I'm going to be much more suspicious of you than if you do. Of course, if you do have a genuine reputation for this then then you're only going to get to exploit that reputation once; as soon as you use that reputation to trick someone it's gone for good.
Additionally, that reputation is actually worth something. In a two-player zero-sum game it's meaningless, but in a three-player interaction it might be beneficial for you to actually convince me that you're being random, because there are situations where your being random and my knowing that you're being random benefits both you and me at the expense of the third player. Sure, but what about 99%? 80%? 50%? If the threat is a huge one, then even if your opponent only has a 10% chance of following up on that threat it should already weigh heavily on your mind. If you assume that a "game" always consists of only a single action by a player and a single action by the opponent, sure. But in general games do not necessarily look like that; many games involve players taking several consecutive actions. Do you think I don't know this stuff?
If we're talking about games where the player takes multiple actions, the finitely repeated prisoner's dilemma is a much better analogue than the infinitely repeated prisoner's dilemma. Let's say, for example, that you know you're going to play 100 rounds of Prisoner's Dilemma, and then the game ends. The interesting thing here is that the game-theoretic "rational" strategy is to defect every round, for 100 rounds.
It seems to me that the point about randomization, and the point about your threats necessarily not being credible, are in conflict with one another.
If there's a situation where the Nash equilibrium would involve randomization, there is no need for you to actually randomize. After all, not being random is only a concern if your opponent can deduce what action you took. As such, you only need to randomize if there's a concern that your opponent may be able to predict you.
So, let's say your opponent can predict you in some way. For example, you have some kind of "tell" that they may be able to use to figure out your move. If that's the case, why is it that they are necessarily unable to predict that you will, in fact, actually follow through with your threat?
The first scenario posits that human beings are at least somewhat transparent, whereas the second one posits that human beings are completely opaque.
Very closely related: Newcomb's problem. The two-boxer's argument is that you can't credibly one-box, because the contents of the box are predetermined, and hence you will always do better by two-boxing. Yet the one-boxer walks away with a million dollars, and the two-boxer only gets $1000.
"If you're randomizing I don't see how it really matters whether your opponent knows that you're randomizing or not." See the section of the video where this is covered, using Netrunner as an example.
"As for "bluffing" in general, we were talking about a scenario where you want to credibly precommit to taking some kind of action." Then this is no longer bluffing. See section in the video about playing chicken.
"If they just wanted to be unpredictable all they had to do was pick randomly; there is no need for them to let me see it." See the section in the video about a bot playing against you at rock paper scissors.
"As such, there has to be an additional reason why they wanted me to see their display of apparent randomness, and I'm obviously going to spend at least some time thinking about what that reason is." The reason is explained in the video. Also, if you show true randomness (like with the dice in Netrunner) you are actually helping the game move more quickly by removing that deliberation.
"Nope, reputation still matters here." Reputation is just another factor to calculate that is bypassed by fake randomization, and can go either way.
Let me break this down.
Player 1 is making a decision.
Player 2 has to respond.
Course of action A
Player 1 estimates that decision X is better against player 2 than Y, say 75 damage vs 25 damage.
Player 1 is actually correct.
Player 2 thinks that player one will make a 50 50 estimate.
Player 1 flips coins to decide between X and Y.
Player 1 abides by the randomness, and goes with X.
Player 2 believes player 1 stuck with the coin flip.
Player 2 will respond to block X, not Y, because it is just as likely as Y but more damaging.
Course of action B
Player 1 estimates that decision X is better against player 2 than Y, say 75 damage vs 25 damage.
Player 1 is actually correct.
Player 2 thinks that player one will make a 50 50 estimate.
Player 1 flips coins to decide between X and Y.
Player 1 abides by the randomness, and goes with Y.
Player 2 believes player 1 stuck with the coin flip.
Player 2 will respond to block X, not Y, because it is just as likely as Y but more damaging.
Player 2 takes 25 damage. Bad, but not as bad as 75.
Course of action C
Player 1 estimates that decision X is better against player 2 than Y, say 75 damage vs 25 damage.
Player 1 is actually correct.
Player 2 thinks that player one will make a 50 50 estimate.
Player 1 flips coins to decide between X and Y.
Player 1 ignores the coin flip, and picks X.
Player 2 believes player 1 stuck with the coin flip.
Player 2 will respond to block X, not Y, because it is just as likely as Y but more damaging.
Player 2 blocks all damage, saving 75 health, despite the coin flip deciding Y.
Course of action D
Player 1 estimates that decision X is better against player 2 than Y, say 75 damage vs 25 damage.
Player 1 is actually correct.
Player 2 thinks that player one will make a 50 50 estimate.
Player 1 flips coins to decide between X and Y.
Player 1 ignores the coin flip, and picks Y.
Player 2 believes player 1 stuck with the coin flip.
Player 2 will respond to block X, not Y, because it is just as likely as Y but more damaging.
Player 2 takes 25 damage.
This kind of reasoning can continue. You see, the point is that FAKE randomness, of randomness ignored by the player, can supplement the informed bluffing of player 1. This only works when player 1 actually knows something player 2 doesn't know they know, the fact that both attacks are not equal, and that one will do more damage.
So this works out better for player 1. The only way for player 2 to respond properly is to use ACTUAL randomness. And then the best course of action by player 1 knowing that player 2 will respond randomly is to always play X, because that will have the same chance of working but deal more damage. But then if they do that, without showing randomness, then player 2 will forget the randomness angle, and always block X, because that will be the obvious thing...
Don't you see? The only way for the two players to stop second guessing each other is for one to unambiguously pre-commit! For example, instead of flipping a coin and then picking a card from their own hand, just hold on the two cards, have the other player flip a coin, and let the other player pick one of the two cards. Then lay the card on the table. Then flip it over when the time comes to reveal the decision that has already been made.
ALL reputation and ALL bluffing and ALL every other consideration can be negated by such pre-commitment, whether the final choice is random or not.
"Sure, but what about 99%? 80%? 50%?" Did you see any of the many parts of the video where they talk about heuristics?
"If we're talking about games where the player takes multiple actions" That doesn't matter. Every single instance of rock paper scissors is utterly divorced from every previous instance for all eternity. Unless we are discussing a specific game there is no point in going further with your wishy washy language and examples.
That's enough for now. Watch the video all the way through, and then see if you've got something concrete to add or dispute.
Perhaps we're talking past each other due to semantic differences, or perhaps one of us isn't understanding something. If it's the latter, I'd bet good money that it's you and not me. If you're capable of putting on a display of fake randomness then the same game-theoretic principles that make your threat non-credible also imply that your bluff is simply just cheap talk. Yes, I saw it. All of their arguments in that section were reasons to randomize, and to use an external device to avoid human error. However, not a single one of those arguments gave a reason why the opponent needs to know that you're randomizing. I said "want to credibly precommit", in the sense that if you *could* credibly precommit you would choose to do so. I didn't say it was necessarily possible for you to precommit. Yes, if you really need to be genuinely random you should use an external device. However, it doesn't matter whether or not I know that you're being random. I saw an explanation for being random. I didn't see an explanation for putting on a show of being random. What is it?
So, to start with, you are correct about one thing. There is no way to avoid the continuing chain of "second-guessing" without randomization. However, as I will demonstrate later, there is no need for the other player to know your'e randomizing. You can randomize in secret, and it achieves the exact same thing.
On a side note, you've assumed that the chain necessarily goes on forever. It's entirely possible that you're capable of outpredicting your opponent, and then you don't need to randomize. Take, for example, the example of the rock-paper-scissors bot that predicts human patterns. If you're playing against that bot, I agree; your best strategy will likely be to play randomly with some external randomization device. However, let's look at the situation the other way around. If you're that bot, with all of its capabilities, and you're playing against a human, being random is obviously stupid. If you can reliably outpredict humans, why would you throw that capacity away by choosing to be random instead. General point: if you are smart enough to outpredict your opponents, you may not need to be random.
OK, now let's consider a case where you're player 2 really is smarter than you, and is capable of following whatever chain of reasoning you follow in order to decide upon a move. In that situation you definitely do need to randomize. Here's what you can do:
P1's strategy: Choose X 25% of the time, and Y 75% of the time.
You could pretty easily do this by flipping a coin twice, and only playing X if the coin comes up heads both times (if you do this, make sure your opponent doesn't see how the coin came up).
Now, if your opponent always blocks X, then 75% of the time they take 25 damage, and so you do an average of 18.75 damage. If your opponent always blocks Y, then 25% of the time they take 75 damage, and so you do an average of 18.75 damage. If your opponent flips a coin, guess what - still 18.75 damage on average.
At this point, you might say "Ah, but what if player 2 is stupid and always plays Y? Obviously player 1 could do better by playing X in that case."
Well, you have a point; if player 1 can work out that player 2 is stupid and out-guess them, then player 2 will win. However, if player 2 is worried that player 1 might outwit them, then player 2 can do this:
P2's strategy: Block X 75% of the time, and Y 25% of the time.
Now, regardless of P1's strategy, P2 is going to take an average of 18.75 damage.
Guess what? We've broken the chain of second-guessing, and you didn't have to make any kind of precommitment to do it! This is the whole point of a Nash equilibrium. If both players intend to play a Nash equilibrium strategy, neither player will have any incentive to switch. No more infinite guessing, just two players randomizing on their own. Neither player needs to precommit; neither player needs to make a visible show of the fact that they're randomizing; it makes no difference whatsoever. Either way, on average, player 2 is going to take 18.75 damage.
Huh? Did you totally miss the point? Games that only involve a single action, like rock-paper-scissors or the Prisoner's Dilemma, are not the usual case. In most games you are able to take several actions in sequence.
In such a game, you can easily have more than one opportunity to follow through with a threat that would, under game-theoretic terms, be "non-credible" (because in following through with it you are hurting yourself).
Besides that, it seems arbitrary to exclude reputational effects that carry over from outside the game. After all, when you're playing with other human beings you carry over your own knowledge of those people, don't you? Hell, even the fact that your opponent is a human being and not a game-theoretic "rational agent" is already enough to cause major issues as far as pure game theory is concerned.
No. Not at all. Stop thinking that. You are the one who brought up.
I said "In a single game with no iteration, there is no way for an opponent to know your track record."
You said "Besides, you can even establish a reputation of this sort within the context of a single game. If you take multiple opportunities to make a threat..."
So I said "Second, I specifically said "In a single game with no iteration, there is no way for an opponent to know your track record." You can't now say "If you take multiple opportunities to make a threat, and on the first time follow through with that threat"... you've simply taken the argument outside the scope of my point, and the point of the argument presented in the video."
And now you've said "On a side note, you've assumed that the chain necessarily goes on forever."
No I haven't. You obviously think I'm saying something more than I'm saying. I'm not. I'm talking about one decision made one time only. Ever. Once. Only once. Not iterated.
I'm not going to cycle through this any more unless you understand.
Player 1 KNOWS that he has two choices. He KNOWS that X will cause 75 damage and Y will cause 25 if successful.
Player 1 KNOWS that player 2 DOESN'T know that player 1 knows this. Player 2 thinks player 1 thinks both choices are equal.
Read that again. It's a key line.
Player 1 PRETENDS to use chance to pick between X and Y. He does this TO CONFIRM to player 2 that he thinks X and Y are equal. But really he doesn't.
Player 1 ignores the chance, and picks Y.
Now it's player 2's turn.
Player 2 thinks player 1 thought both X and Y are equal. He thinks this because he didn't know player 1 kept track of the cards, or found in some other way, in advance.
That player 1 thought both cards are equal is NOT TRUE but, due to the flip of the coin, this LIE is confirmed in the mind of player 2.
Because, in player 2's mind, player 1 has flipped a coin to decide between X and Y, they will block against the more damaging attack 100% of the time. Player 1 has already played. They can't change their mind, nor can they play again in the future. There is no future. There is only now.
Player 2 blocks X.
Player 2 takes 25 damage.
This does not work if player 2 doesn't "know" that player 1 used a coin flip to decide X. This doesn't work if player 1 hides the fact that they are "using" random chance.
But this is the entire point. This is the answer to your question. It is bluff using fake randomness:
"If someone makes a blatant, visible display of apparent randomness then that's an important sign. If they just wanted to be unpredictable all they had to do was pick randomly; there is no need for them to let me see it. As such, there has to be an additional reason why they wanted me to see their display of apparent randomness, and I'm obviously going to spend at least some time thinking about what that reason is."
And again, this only works when one player is convinced that the player is committing to the random play. Pretending to shuffle two cards and pick one is good enough for most opponents. But the deception is the key.
The only way you should ever consider any show of randomness is if it is locked in. Pre-committed. This is exactly what Rym and Scott are talking about in the video. Iterations are not needed.
I'm pretty sure that's what you meant by "only way the players can stop second-guessing each other"---or did you mean something totally different?
OK, it looks like I've just misunderstood your usage of "iterated", because you used it in a non-standard way. An iterated game is a bigger game where you play the exact same smaller game (the stage game) over an over again. See this Wikipedia article for a reference.
A game where you only make a single decision is a different question, but so what? If Rym and Scott are telling you how to win "Every Game", then it's still true that the vast majority of games are not like that. In the context of a longer sequential game you most definitely could have multiple opportunities to make threats; sure, they might be different threats in different situations, but the point is very much worth discussing.
Even if we assume Player 2 is not 100% convinced that Player 1 thinks both choices are equal, the show of randomness doesn't really help. If I'm Player 2 and I see Player 1 putting on a show of randomness, I will be more suspicious, not less suspicious.
If you are more suspicious, then you are not player 2. You are player 1. You already know shows of randomness are only 100% not bluffs when 100% pre-committed. As in, throw the steering wheel out the car.
Again, you can use the show of randomness as a trick to fool less competent players. This is one tool in how to win every game. You can ignore ALL shows of randomness and bluffing and EVERYTHING ELSE outside the game mechanics, UNLESS the action is 100% pre-committed. Then you have to believe it. This is how you not-lose every game.
Unless it is iterated, and it's the same thing over and over.
But I specifically said NOT that. Not that. Different moments, or turns, or choices, in a single non-iterative game must be approached equally. Threats don't carry forward when there is a change in game states.
This is the reason why the finitely repeated prisoner's dilemma breaks down. Threats don't carry over, but your knowledge of your opponent does. If your opponent has carried out a threat that game theory tells you is non-credible once before, twice before, three times before, what do you do the fourth time? Yes, it's not the same threat, but it's still the same opponent.
Among human players in games, I find that all intentional signaling is literally worthless. If a "threat" is made, that action that is threatened was already taken into account as an option by both players. The threatening player's choice to follow through is either a good move for him or not, irrespective of my reaction. If I alter my behavior based on the "threat," I should have made that same alteration without it.
The fact that the option exists at all implies a threat regardless of intentional signaling.
By reacting to any intentional signal in any practical game, I only give power to the person injecting this otherwise meaningless signaling. If I back off because of a signaled threat, then the threatener has gained a boon at no cost to himself.
Following through on a threat costs resources of some kind (or at least opportunity cost). If it does not, then the threatened action will (should) be undertaken regardless of my reaction as it is objectively superior to other actions.
Threats are only irrevocable past the point of input into the game. They are meaningless before that. Until I play my card, or let go of my piece, all of my actions are theoretical, and all of my signals are bluster atop already known quantities.
If you respond to intentionally signaled threats in a game ever, so threatening you becomes a permanent meta strategy. By never responding to a signaled threat ever, you force your opponent to either harm themselves over time to achieve nothing (maintaining their "reputation") or give up on threatening you.
Not responding to threats wins over always (even to one's detriment) following through on them in repeat play.
But I think we're having different conversations here. Do you want to play a perfect robot game machine? Or do you want to play a human? Do you want to play a human who is better than you, or worse than you?
If I'm playing games with people at my level I shit-talk constantly, and make threats all the time. They all do it to me too. But you know what? It's just background noise. We all know it. In the end it comes down to skill. The background noise is what makes it a social activity, not a solo computer simulation.
And if you are playing to win, ignore all threats. Only ever play with the options you have with the knowldge you have. All other out-of-game signalling by any other player is dubious.