Three-player 0-1 poker w/ redraw

Please look into the three-player-lottery-elimination thread for reference. This here is a variation.

Three players each receive a random number between 0 and 1. The goal is to end up highest.
First player has the opportunity to keep his number or redraw. This choice is common knowledge.
Then Player 2 has the same option (knowing what 1 did). Then Player 3 ditto.

I computed the levels under which to redraw in each situation : P1, P2 if 1 kept, P2 if 1 drew, P3 if both kept, etc. (7 in all)

What would you do as Player 1 ?
POLL: who do you think has the advantage?

If the redraw were not public, it would be symmetric between all players. Since they are public, Obviously player 3 has the advantage to know more about the game state than any other player.

Quote: MangoJ

If the redraw were not public, it would be symmetric between all players. Since they are public, Obviously player 3 has the advantage to know more about the game state than any other player.

Yep, that's the easy part of the question for sure.

And for the other part, I'm going to take a wild guess at patting with higher than 0.8; not in the mood to try the math right now.

why is I am a bigot not listed as a poll option ? this is heresy.

Quote: MangoJ

If the redraw were not public, it would be symmetric between all players. Since they are public, Obviously player 3 has the advantage to know more about the game state than any other player.

In fact, this argument just shows that Player 3 does better than he would if he had no info. It does not guarantee that 3 has the overall advantage.

Quote: kubikulann

In fact, this argument just shows that Player 3 does better than he would if he had no info. It does not guarantee that 3 has the overall advantage.

I'm not saying player 3 has the advantage solely because he has more information.
I'm also saying that without public information, the game is equal among all 3 players.

Since more information about the game will not reduce your advantage (you could always choose to ignore it), it will either increase your advantage or have no effect at all, but it will never have a negative effect.

It's not a prove that player 3 has a non-zero advantage, they could all be equal in the end. But if there is a player with the highest advantage, it will be player 3.

Quote:

Spoiler

This reads as a challange. Luckily it's weekend, I might actually give it a try to solve this game.

In games of incomplete information, the advantage does not always go to the last one.
Take into account the possibility of previous players to act in a way that manipulates (partly) the actions of the next ones through the information they willingly reveal.

In fact, that's the essence of bluff in poker...

Here there is no bluff per se (since no betting), but you still have a Stackelberg*-like situation. (*Asymmetric info game in economic theory).

I voted that the third player has the advantage. This would be a tough analysis. I'd like to suggest we discuss the two-player case first.

Alright, it seems the interest has waned. Let's write the solution.

Actually, if all players play optimally (in the sense of Nash), then the advantaged player is #2.
Probabilities of winning are : (#1) 28.85% , (#2) 38.79% , (#3) 32.36% .
(If I didn't boggle the calculations. Lots of integrals. A doublecheck is welcome.)

I guess this comes from the fact that 2 and 3 have info on what previous players did, but 1 and 2 have the opportunity of 'signalling', i.e. influencing the later players through manipulation of the info they choose to give. Probably #2 benefits from having both?

Strategies are :
#1) redraw if you get less than 0.646677
#2) if #1 kept, then redraw if less than 0.765705 ; if #1 redrew, redraw if less than 0.616876
#3) if #1/#2 ... then redraw if less than ...
-- keep/keep -- 0.811529
-- keep/draw -- 0.722857
-- draw/keep -- 0.707378
-- draw/draw -- 0.577350 (= sqroot of 1/3)

Quote: kubikulann

In games of incomplete information, the advantage does not always go to the last one.
Take into account the possibility of previous players to act in a way that manipulates (partly) the actions of the next ones through the information they willingly reveal.

In fact, that's the essence of bluff in poker...

You can always choose to ignore such information. Hence the last player can never perform worse (if the game is otherwise equal).

Quote: MangoJ

You can always choose to ignore such information. Hence the last player can never perform worse (if the game is otherwise equal).

I can see the relevance of that. But it bothers me, in that it seems to imply that the last player in a poker round always has the advantage? Maybe it's the case; I'm not a poker player. What do you reckon?

Quote: kubikulann

I can see the relevance of that. But it bothers me, in that it seems to imply that the last player in a poker round always has the advantage? Maybe it's the case; I'm not a poker player. What do you reckon?

Yes, on a given hand the last player to act (if everybody checks) is the "button", and is considered by far the strongest playing position. (Technically only in the pre-flop betting round he is not last to act, the small blind and big blind position may act afterwards - but after that he is the last to act).

However, between all hands the seating positions rotate thus making it fair on cash-games.

Quote: kubikulann

I can see the relevance of that. But it bothers me, in that it seems to imply that the last player in a poker round always has the advantage? Maybe it's the case; I'm not a poker player. What do you reckon?

Yeah, I'm bothered by the results too. Last player has the most information. He should be winning the most I would think.

Quote: tringlomane

Yeah, I'm bothered by the results too. Last player has the most information. He should be winning the most I would think.

Let us imagine a simpler setting of the game: only six cards to draw, no replacement.

In this case, the third player may benefit from more information, but this is possibly compensated by the reduced card set he is presented?
I'm not at all sure of that. Just thinking out loud...

Could it simply be that the solution is not correct ? Maybe the Nash strategy does not consist of a single threshold ?

If P3 doesn't know the choice of P2, he will have same equity as P2. Hence he can always employ the same strategy P2 and will have greater equity..

Quote: MangoJ

Could it simply be that the solution is not correct ?

Might be !...
A simulation yields percentages of wins close to (1) 30% (2) 34% (3) 36%.

Shame on me.