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Author Topic: First player bias  (Read 38059 times)

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timchen

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Re: First player bias
« Reply #100 on: May 03, 2012, 01:18:47 am »
+1

No, you shouldn't. The game would end, and you would not improve your position by buying it. Your correct move is to not buy it. If you do buy it, it's a strategic blunder (possibly from a situation in which your chances of winning were effectively zero), but it's hardly kingmaking, at least in my view.
This is paradoxical. You don't improve your position so you don't buy it. But as I have supposed, you don't have the chance to win if you don't buy it either. So you can't improve your position by not buying it. Why is the correct play to prolong the game and just wait for the inevitable?

The point is that in games with any number of players there are situations where your play just do not affect yourself that much. In a game with more than three players, however, that play can affect the other players much more than it affects yourself. In this case the other players have no way to control their fate; who wins depends on your decision which is irrelevant to yourself.

As others have said, if players don't know each other and cannot communicate, then it is equivalent to some form of luck, for the rest of the players. But otherwise this situation is highly susceptible to collusion.

Quote
Neither. You should try to win, or barring that, get second.
Either strategy can get you to a win. And in this example it is indeed likely that whatever strategy you choose that strategy will win out. So your choice entirely depends on which other player you'd like to help.

Quote
Okay, if kingmaking just means "taking actions that affect other players disproportionately," than there's nothing wrong with kingmaking.
There is nothing wrong as long as you are playing to win yourself. The problem is there are situations how you play does not affect your own winning chance, but still affect others disproportionately.

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Robz888

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Re: First player bias
« Reply #101 on: May 03, 2012, 01:32:14 am »
0

No, you shouldn't. The game would end, and you would not improve your position by buying it. Your correct move is to not buy it. If you do buy it, it's a strategic blunder (possibly from a situation in which your chances of winning were effectively zero), but it's hardly kingmaking, at least in my view.
This is paradoxical. You don't improve your position so you don't buy it. But as I have supposed, you don't have the chance to win if you don't buy it either. So you can't improve your position by not buying it. Why is the correct play to prolong the game and just wait for the inevitable?

Okay, you absolutely, positively will not improve your position if you buy it. You know this for fact. If you do not buy it, your opponents could blunder horribly, and you might improve your position. It could be that you lose either way. Yeah, let's say there are not enough points left for you to improve your standing. Should you buy it? I mean, I would still say no, just in case you overlooked some Curse-giver, or Saboteur, or something. Anyway, I think that situation is rare. In most cases, buying it or not buying it will be better or worse for you, and you base it on that.

Quote
The point is that in games with any number of players there are situations where your play just do not affect yourself that much. In a game with more than three players, however, that play can affect the other players much more than it affects yourself. In this case the other players have no way to control their fate; who wins depends on your decision which is irrelevant to yourself.

Yes, and it's fine to make moves that disproportionately affect certain players. I mean, look, if you go after Possession you really hurt the guy who comes after you, and not the other guy. And if you buy Possession because your thinking, "I want to screw this guy over big!" then yeah, you have a problem, whatever. But you probably bought Possession because you thought it would be best for your position in the game. So the player after you suffers more than the other guy, but it's still strategic, not kingmaking.


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Either strategy can get you to a win. And in this example it is indeed likely that whatever strategy you choose that strategy will win out. So your choice entirely depends on which other player you'd like to help.

Your choice depends on which strategy you think has a .000001% chance of beating the other strategy.

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There is nothing wrong as long as you are playing to win yourself. The problem is there are situations how you play does not affect your own winning chance, but still affect others disproportionately.

I agree that if people are not playing to get their best possible finishing place, you will have problems. But if everyone is playing to win, most situations do affect your own chance of winning, perhaps slightly, but they affect it.
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Robz888

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Re: First player bias
« Reply #102 on: May 03, 2012, 01:36:15 am »
0

Maybe part of the issue is that people tend to give up in Dominion too easily. They think "that's it, I've lost for sure." And perhaps that's why you worry about kingmaking in 3+ players, because you'll have some people who aren't in it. Well, those people need to be careful. They are often more in it than they think. This happens in 2 player, too. I mean has everyone seen this game?

Comebacks happen in Dominion. Or you might be totally off about your odds of winning. You might get a lucky hit with Saboteur. It happens! So if you play to win, even where you seem totally out of it, there's not going to be much of a kingmaking problem.
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timchen

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Re: First player bias
« Reply #103 on: May 03, 2012, 01:54:14 am »
0

No it's not the problem at all. The problem is both a conceptual thing and a practical thing.

While you can continuously speak a percentage of the relevance of your play to your position, at some point this number is just not meaningful. When this number is much smaller than the percentage that you'll affect the relative position rest of players, I'll just ignore it.

And in practice, even if we say, we can rigorously determine this tiny percentage, not everyone plays according to that. And that creates problems. Suppose in a 3p game, first player get $10, second get $5 and third get nothing. If you are in the position that you are third and your decision can improve your winning chance by 1% but you can decide who wins if you choose to do so, why do you not request $3 and play for the player that pays you? Your expected income is much better than 5c, which is what you expect to get if you just play to win. That is collusion.

Sure, further rules can forbid this, but that is not the point of the discussion. My point is to point out there is no such situations in 2 player games. And that is conceptually how 2p games and multiplayer games are different.
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WanderingWinder

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Re: First player bias
« Reply #104 on: May 03, 2012, 02:02:38 am »
0

No it's not the problem at all. The problem is both a conceptual thing and a practical thing.

While you can continuously speak a percentage of the relevance of your play to your position, at some point this number is just not meaningful. When this number is much smaller than the percentage that you'll affect the relative position rest of players, I'll just ignore it.

And in practice, even if we say, we can rigorously determine this tiny percentage, not everyone plays according to that. And that creates problems. Suppose in a 3p game, first player get $10, second get $5 and third get nothing. If you are in the position that you are third and your decision can improve your winning chance by 1% but you can decide who wins if you choose to do so, why do you not request $3 and play for the player that pays you? Your expected income is much better than 5c, which is what you expect to get if you just play to win. That is collusion.

Sure, further rules can forbid this, but that is not the point of the discussion. My point is to point out there is no such situations in 2 player games. And that is conceptually how 2p games and multiplayer games are different.
But this is the whole problem. The percentage is not meaningful, so you say. But really, why not? No seriously, why isn't it? You're not playing to win anymore. So long as it's not absolutely 0, it doesn't matter what the percentage is. And of course, you can have the same situation in a 2 player game. "I'll resign right now if you give me 2 bucks (playing 10-0)." It's the exact same thing.

timchen

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Re: First player bias
« Reply #105 on: May 03, 2012, 02:12:44 am »
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Exactly. Except in the 2p game, that player won't give you any. If you are already losing, he can just take the whole $10.

To put it more clearly, in a 2p game, collusion is at your own expense, whereas in a game with more players, it is at others. There is no problem to collude at your own expense (as it is not distinguishable to very incompetent play), but collude at other's expense is totally different.
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Robz888

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Re: First player bias
« Reply #106 on: May 03, 2012, 02:20:06 am »
0

Suppose in a 3p game, first player get $10, second get $5 and third get nothing. If you are in the position that you are third and your decision can improve your winning chance by 1% but you can decide who wins if you choose to do so, why do you not request $3 and play for the player that pays you? Your expected income is much better than 5c, which is what you expect to get if you just play to win. That is collusion.

This scenario violates my one condition: that getting 1st is preferable to getting 2nd, and 2nd to 3rd. In the above situation, getting 3rd is preferable to taking a 1% chance at getting 2nd, because you are colluding and splitting the pot. So yeah, that's bad, and I explicitly ruled that out from consideration when explaining how you could play 3+ Dominion without kingmaking (namely, you can't do the thing you just suggested people could do).
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timchen

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Re: First player bias
« Reply #107 on: May 03, 2012, 02:38:57 am »
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Sure, for that I agree: If you ban collusion there will be no collusion problem.

For that matters, the situation we talked about will then just increase the variance, especially when people have trouble identifying their 1% advantage when making their decisions. All I am saying here is this increase of variance is an intrinsic property of games involving more than two players; and one interesting property of this particular variance is that if you don't ban collusion it will lead to collusion problems, which is entirely absent in 2p games.
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Robz888

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Re: First player bias
« Reply #108 on: May 03, 2012, 02:59:25 am »
0

Sure, for that I agree: If you ban collusion there will be no collusion problem.

For that matters, the situation we talked about will then just increase the variance, especially when people have trouble identifying their 1% advantage when making their decisions. All I am saying here is this increase of variance is an intrinsic property of games involving more than two players; and one interesting property of this particular variance is that if you don't ban collusion it will lead to collusion problems, which is entirely absent in 2p games.

I guess I was talking more about kingmaking than collusion. You could introduce all sorts of outside-game reasons, I guess, for two players to work together, that wouldn't hold if each person was just playing to finish best. As WW said, you can have this in 2 player. Here's $10 if you lose on purpose. That's collusion, it's just collusion that doesn't screw a third party. So anyway, yeah, collusion should be prohibited in both cases.

You know, you could make the argument that collusion is much more possible in 2 player. If I pay you to throw the game, hey that's easy. In three player, I have to pay two people to throw the game. Because even working together only goes so far. Depending on the Kingdom, it's probably more likely than not that you could beat two people who were explicitly working together. Masquerade, Possession, Ambassador, Attacks might trip you up, but yeah... it's not as big a deal as in a lot of other games.

Anyway, if everybody is playing to win, you will not have much of a kingmaking problem in 3+ players, because virtually every decision you make affects whether YOU will win the game or not. Okay?
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timchen

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Re: First player bias
« Reply #109 on: May 03, 2012, 03:31:44 am »
0

No. You are still missing the point. Maybe this example can make it clear.

Suppose in my early example, the player made the "wrong" strategic choice of getting that Duchy.

Suppose you are the second ranked player. Now you are angry about the situation, because you lost due to a wrong play from someone else. You explained to that player in length why he shouldn't take that Duchy. Well, He shrugged and moved on.

It's not a problem of whether this is called collusion or king making. The point is that this phenomenon does not exist in 2p games. In 2p games an error from your opponent can only benefit you.

In your words, if everybody is playing to win, you will have a kingmaking problem in 3+ players, because other players can win by your mistake. And even if your decision is not a mistake, it can take long long time to persuade others that you are right. And frankly speaking, if your correct play only produce 1% advantage for yourself, you shouldn't be optimistic at convincing the other players that you are not kingmaking...

« Last Edit: May 03, 2012, 04:20:26 am by timchen »
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WanderingWinder

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Re: First player bias
« Reply #110 on: May 03, 2012, 08:16:54 am »
0

timchen, you're upset that you can lose because of what someone else does, which makes neither them nor you actually win. This is just a part of the game though. Every multiplayer game can and does have that, bad play or not. It's fine that you hate it, of course, but it doesn't need to be nefarious.
And my earlier point is that the 2 bucks can happen in 2 player also. I only have a 10% chance of winning right now, or something terribly low (I can't likely calculate precisely), so I look to make the deal. Obviously if it's exactly 0, they reject. Hey, that's also true in multiplayer.

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Re: First player bias
« Reply #111 on: May 03, 2012, 08:30:09 am »
0

In 2p against an optimal opponent, you can't improve your expected value with a deal.

If you offer too much (more than reward * p to win), your opponent rejects.

If you offer too little (less than reward * p to win), your opponent accepts, but you would have been better off playing it out.
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WanderingWinder

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Re: First player bias
« Reply #112 on: May 03, 2012, 08:33:29 am »
0

In 2p against an optimal opponent, you can't improve your expected value with a deal.

If you offer too much (more than reward * p to win), your opponent rejects.

If you offer too little (less than reward * p to win), your opponent accepts, but you would have been better off playing it out.
Right. You can abate your risk though.

In 3p, you can't improve your expected value with a deal either (that both other players accept). To do it without the consent of everyone else, that's going to be totally illegal.

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Re: First player bias
« Reply #113 on: May 03, 2012, 08:42:34 am »
+2

Surely the colluders agree apriori.

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WanderingWinder

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Re: First player bias
« Reply #114 on: May 03, 2012, 08:49:43 am »
0

Surely the colluders agree apriori.

In which case we're talking about game-throwing. Which is also a problem in 2 player.

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Re: First player bias
« Reply #115 on: May 03, 2012, 09:00:56 am »
0

The difference is that it is mutually beneficial to collude.  We can both win more at the expense of the third player.
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Re: First player bias
« Reply #116 on: May 03, 2012, 09:03:30 am »
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The difference is that it is mutually beneficial to collude.  We can both win more at the expense of the third player.
No. It's not mutually beneficial. Only one player is benefitting....

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Re: First player bias
« Reply #117 on: May 03, 2012, 09:10:50 am »
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Imagine there is a set where there is a rush strategy that will win if 2 of 3 players go for it, and lose if only 1 of 3 players goes for it.

I agree with you that we'll both go for it iff the 3rd player does not.
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Re: First player bias
« Reply #118 on: May 03, 2012, 09:16:54 am »
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Imagine there is a set where there is a rush strategy that will win if 2 of 3 players go for it, and lose if only 1 of 3 players goes for it.

I agree with you that we'll both go for it iff the 3rd player does not.
First of all, this is a gross oversimplification of any kind of game state. But moreover...
First player shouldn't do this, because he's likely to win anyway, going with the non-rush strategy. So he's actually hurting his chances. Now, second player maybe should or maybe shouldn't, depending on the amount of help he needs, and how good his chances are in the non-rush. Third player obviously goes after it iff one of the other two has. And because you can expect that to happen, you keep that in mind when choosing your strat as p1 and p2. There is a prior equilibrium, game-theory wise. And disturbing that equilibrium is, in any event, going to hurt that person who disturbs it. Not foolproofedly, but in the vast vast majority of cases.

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Re: First player bias
« Reply #119 on: May 03, 2012, 09:31:16 am »
+1

I agree it's a simplification.  Really what happens is that there is a rush strategy that is not quite viable alone (p = .25, say) , but when paired with another rusher, beats the long game strategy (p = .75 combined, perhaps).  I am sure you can find some kind of weak gardens strategy that has this property with the simulator if you tried.  This also requires that players strongly and irrevocably commit to different strategies, which isn't totally true in dominion either.

Of course, acting in concert requires trust and coordination.  But given that you trust your co-conspirator, you can definitely both get ahead by 'playing for the team'.
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WanderingWinder

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Re: First player bias
« Reply #120 on: May 03, 2012, 10:02:23 am »
+1

I agree it's a simplification.  Really what happens is that there is a rush strategy that is not quite viable alone (p = .25, say) , but when paired with another rusher, beats the long game strategy (p = .75 combined, perhaps).  I am sure you can find some kind of weak gardens strategy that has this property with the simulator if you tried.  This also requires that players strongly and irrevocably commit to different strategies, which isn't totally true in dominion either.

Of course, acting in concert requires trust and coordination.  But given that you trust your co-conspirator, you can definitely both get ahead by 'playing for the team'.
This doesn't address my basic point that: either one of them has to decrease his/her own winning chances in order to do this, in which case it is, to some extent, throwing, and definitely not mutually beneficial, or it helps them both, in which case it's probably just good strategy rather than collusion. Or it's collusion but the effect isn't different from good strategy. Either way, I'm not seeing the problem, even in this unrealistic, super-simplified scenario. Unless the problem is throwing. 'cause yeah, that's a problem.

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Re: First player bias
« Reply #121 on: May 03, 2012, 10:10:03 am »
0

A Bureaucrat/Gardens strategy vs BM Smithy could lead to interesting observations in multiplayer. Playing a bit with the simulator I found that if one player goes for Smithy, the three other players can gang up by going Bureaucrat/Gardens and beat Smithy. If one player goes for Bureaucrat/Gardens the others can all go Smithy and beat him. But the bots are far from optimized and this result might be wrong.
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Re: First player bias
« Reply #122 on: May 03, 2012, 10:11:19 am »
0

A Bureaucrat/Gardens strategy vs BM Smithy could lead to interesting observations in multiplayer. Playing a bit with the simulator I found that if one player goes for Smithy, the three other players can gang up by going Bureaucrat/Gardens and beat Smithy. If one player goes for Bureaucrat/Gardens the others can all go Smithy and beat him. But the bots are far from optimized and this result might be wrong.
Yeah, but this isn't collusion. It's just strategy.

timchen

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Re: First player bias
« Reply #123 on: May 03, 2012, 10:27:45 am »
0

... I can only say, when someone refuses to see something, it cannot be helped. But maybe just let me try one last time:

In my previous Duchy example, suppose I can improve my position by 1%, not buying the Duchy.
If the prizes are $10, $5, and $0, I can reasonably request $2.5, if I am mean I can request arbitrarily close to $5 from either player. The amount I can request has nothing to do with my position.

If there are only two players, again suppose I can improve my position by 1%...
If the prizes are $10 and $0, now I can only request 20c.  The amount I can request is just proportional to how much my position is changed.

That's how the two kinds of collusion are different; the second one is not really collusion as you effectively trade nothing.
« Last Edit: May 03, 2012, 10:33:46 am by timchen »
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Re: First player bias
« Reply #124 on: May 03, 2012, 10:40:14 am »
0

... I can only say, when someone refuses to see something, it cannot be helped. But maybe just let me try one last time:

In my previous Duchy example, suppose I can improve my position by 1%, not buying the Duchy.
If the prizes are $10, $5, and $0, I can reasonably request $2.5, if I am mean I can request arbitrarily close to $5 from either player. The amount I can request has nothing to do with my position.

If there are only two players, again suppose I can improve my position by 1%...
If the prizes are $10 and $0, now I can only request 20c.  The amount I can request is just proportional to how much my position is changed.

That's how the two kinds of collusion are different; the second one is not really collusion as you effectively trade nothing.
No, I get this. But you can't do the second thing, because it's absolutely illegal. If you could, sure, it would be a problem.
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