Or look at it this way. I'm playing the Devil heads up. I win I go upstairs, I lose I burn in hell. We have one chip and there are no antes. We can bet or fold.

He gets AA and bets. I get KK and call. His hand stands up. He gives me a second chance and we start over.

This time we play with the hole cards face up. He gets AA and I get KK. He bets. I fold, I keep my chip and he takes the pot (his chip).
We deal again face up. I get KK and he gets AA. I fold.
We deal again.............

I'm a patient type but 10 years down the line, I go with my hand.....

Just because you lose money on a hand in total doesn't mean you can't make a +EV decision on a single street.

That's what I've been saying. But a single street has at least two separate decisions.

if you are in a spot where both players can make +ev decisions then that means one of them made a -ev decision on an earlier street

HU 2 players:

There is 1.5BB in the pot preflop before any betting is done.

Both players have AA, cards turned up.

Its Ev+ for both players to go all-in preflop.

You are forgetting the blinds.

Paying the blinds is always a EV- decision but its forced and you can do nothing about it.

They can check-check and still be 50-50 so open shoving is not a +EV decision. If first to act goes all in the the second player is +EV to call. They are never +EV at the same time.

Maybe I'm wrong. If the two players have face up AA (or 22 or 97 with all four suits showing etc etc) then is it +EV for the SB to fold because otherwise it is playing to chop a pot the that the BB can chop for free?

The thing that is "summed" in a zero sum game isn't "EV" it's winnings and losings.

I agree that if an unknown entity dumps money in the pot that it's no longer a zero sum game, if you are somehow allowed to ignore the source of the money. I mean, it's not zero-sum in respect to the "system" the game is played under, clearly.

I would say one of the players got the other player's range wrong.

^ No, not necessarily. If we talk about ranges both players can be +EV. I ship KK anyday against somebody openshoving every hand. If he's openshoving AA he obviously made a +EV move but me calling is +EV aswell.

Say HU there's 1bb of antes in the pot and no blinds. You have AA. Your opponent shoves, and shows AA. You call. You are in a neutral ev spot. However calling was +ev because calling is better than folding.

Yep, and you calling is -EV for your opponent. Zero sum.

The phrase zero sum game comes from the study of game theory. Game theory came from studying poker players and then got applied to all games, then political and social sciences. So poker is basically one of the first recognized zero sum games.

In this case, the pot is 1bb + 2 * Bet. Each player having aces has an EV of 0.5bb - they both have positive EV.

At showdown, after all bets are made and board has been dealt - one player will win and one player will lose what is in the pot. Maybe you are confusing that situation with players having EV now with future betting and/or cards to be dealt.

My opponent's decisions do not affect my EV.

No, you are confusing two actions at different times with two simultaneous actions. Your opponent shoves and shows AA. You have AA. You calling is +EV for you and -EV for your opponent. At showdown yes one player might win and one player might lose - but this is a long shot, it will probably be chopped. Also, it doesn't have to be AA vs AA. It could be 55 vs 55, 8d4c vs 8s4h - the mathematics is the same. Only a flush prevents a chop.

If you can see your opponents hole cards and you both have 22 then you will shove because your opponent folding is +EV for you. Your opponent then has a decision to make, you already made yours. Card face down your opponent can fold 22 vs 22, but can't fold AA vs AA. Face up, the correct EV decision is the same regardless of the 50-50 match up.

Do you really understand what EV is? It is the average of how much one may win or lose if the same situation occurred a number of times. It is a statistical measure that account for outcomes of random events.

The EV of the player who shoved with aces depends on four things
1. How much is in the pot
2. How much he will invest
3. Whether his opponent will call or fold
4. What cards will fall on future rounds

Now with aces preflop the EV of a shove HU is clearly positive. If his opponent folds he wins X, say. If villain calls, he wins X-d, say, a smaller amount on average for he will almost surely split the pot. His EV reduces, but that amount, X-d, the EV for a call by his opponent, is still positive. As I stated earlier, for the simple example of 1bb in the pot, both players have EV of 0.5 bb -- the pot is split.

Summary: For this example, both players have positive EV wrt to their decisions and wrt to that of their opponents. Any other conclusion is pure fantasy







-d, the

To follow up on my previous reply. The EV equation for a bettor who goes all in is the following

EVbet = F*Pot – (1-F)*[eq(Pot + Bet) – (1-eq)*Bet]

The EV for the caller in the same situation is

EVcall = (1-eq)*(Pot+Bet) – eq(Bet),

where

F = probability opponent folds to your shove
Pot = amount in Pot prior to the shove
Bet = amount of shove
eq = probability you will win the hand, which is the same as opponent losing the hand

Now if Pot is large enough, you can show that both EVs > 0 as one example where each of two player’s decisions can be +EV

For the example posed earlier of each player having aces, eq = 0.5 and F= 0 (opponent always calls). If you put these values in the equation you get

EVbet = Pot/2; EVcall = Pot/2, both positive.

To say this cannot happen seems to me that you are using another interpretation of the meaning of two players having +EV.

The only time poker isn't zero sum is when someone hits a BBJ.

Of course both decisions can be +EV. But they are not simultaneous.

Here is OP's statement. On any hand the players make decisions - they obviously are not simultaneous if you mean at the same instant they decide what to do.

Just what to you mean by a simultaneous decision relative to OP's statement?

Poker is a zero-sum game, not including rake.

However, it is possible for two players to make a a +EV decision in the course of a hand. In fact it comes up quite often.

For instance, let's say there is $10 in the pot and each player has 5$ left. On the flop one player flops top pair and the other flops a flush draw.

Well, the first player says "I has de best hand, allllllll in"
The second player says "I has getting 3-1 odds, EZ call"

Both players are making +EV decisions BECAUSE they are fighting over $10 that is already in the pot.

It is not relevant if the decisions are "simultaneous". As I mentioned before the thing we "sum" in a zero sum game is not EV. It's dollars won and lost. It is impossible for 2 people to both profit in HU poker, unless you resort to an example where some 3rd party is injecting money into the pot.


(The BBJ answer is cute but false, BBJ pays out less than it takes in)

It is relevant because player A acts first and makes (hopefully for player A) a +EV decision. Player B then makes a +EV decision based on all the actions up to and including player B's last action. Player A's EV has now changed because it no longer depends on the opponents possible actions.

I agree the "sum" is the dollars or chips or whatever and not the EV, but the "relative" EV that you mentioned earlier is zero sum at any given point in time. This is why the non-simultaneity of events is relevant with regard to the OP's statement.

I guess it's still a definition problem.