chase the ace

Hi this is my first question so I hope it's' not misplaces.

In a heads up game of Chase the Ace, what is the advantage to the dealer?
I have tried to calculate this with combinatorics but I cant' solve it.

The dealer D and the non-dealer N each get a card which they look at from the standard shuffled deck.
The idea is to get the highest card and Ace is low.

If N likes his card he can keep it, if he doesn't he can swap with the dealer. Unless D has a King because King;s can't be swapped in the rules I played.
If D likes his card which he may have just been given he keeps it. otherwise he can swap it with the card from the top of the deck.

Obviously there is some strategy. If N keeps his card it will certainly on average higher than 7 the average may be about T or J. So as dealer facing this we need a good strategy. N also need to optimize his keep/swap strategy.

I'm looking for win lose draw percentages or a way to try to get them depending on what strategy the players use.
Do I need a program to run a hundred thousand iterations for each keep and swap strategy for each player?

Grateful for any ideas :-)






I

Look up "Screw Your Neighbor."

Ah.... thanks for that and yes I found it :-) /games/screw-your-neighbor/
But why is the optimal strategy as said.
If the player first to act keeps his card why would the 2nd played choose to stick with 8 or even a 9.
For example, as suggested if the 1st player always swaps 6 or less, a card he keeps will on average a T.

Quote: Spartan62

Ah.... thanks for that and yes I found it :-) /games/screw-your-neighbor/
But why is the optimal strategy as said.
If the player first to act keeps his card why would the 2nd played choose to stick with 8 or even a 9.
For example, as suggested if the 1st player always swaps 6 or less, a card he keeps will on average a T.

The Wizard has the optimal strategy here.

I've seen that but why is that optimal.

If player one always swaps 6 or less which seems fair enough, he has 7-K.
If player two keeps an 8 when player one keeps his..

Then player two will...
Win 4, draw 3, lose 20 against the possible 27 cards player 1 can have Only about 20% win rate.

Won't player one do better if he swaps an 8 ?

Am I missing something here :-)

Quote: Spartan62

If player one always swaps 6 or less which seems fair enough, he has 7-K.

Am I missing something here :-)

(edit - read the post wrong)

Here's what I get:

There is a 4/27 chance of player 1's card being each of 7, 9, 10, J, Q, and K, and 3/27 of it being 8
If it is a 7, then, of the 50 cards remaining, 24 (A-6) lose, 3 (7s), and 23 (3 8s, 4 each of 9-K) win
If it is an 8, then 28 (4 A-6 and 3 7s) lose, 7 (3 8s and 4 Ks - remember, if the top card of the deck is a K, treat it as an 8) tie, and 16 (9-Q) win
If it is an 9, then 35 (4 A-6, 3 7s, 4 8s, 4 Ks) lose, 3 (9s) tie, and 12 (10-Q) win
If it is a 10, then 39 (4 A-6, 3 7s, 4 8-9s, 4 Ks) lose, 3 (10s) tie, and 8 (J-Q) win
If it is a Jack, then 43 (4 A-6, 3 7s, 4 8-10s, 4 Ks) lose, 3 (Js) tie, and 4 (Qs) win
If it is a Queen, then 47 (4 A-6, 3 7s, 4 8-Js, 4 Ks) lose, and 3 (Qs) tie
If it is a King, then all 50 cards lose
The expected value is 4/27 x ((23-24) + (12-35) + (8-39) + (4-43) + (0-47) + (0-50)) / 50 + 3/27 x (16-28) / 50
= (4 x (-191) + 3 x (-12)) / (27 x 50) = -800 / (27 x 50) = -16/27

If the second player stands on 8, then, of the 27 cards player 1 has, 4 win for player 2 (7s) and 20 lose (9-K), so the EV is -16/27.

Looks like there's no difference either way.

Quote: Spartan62

I've seen that but why is that optimal.

If player one always swaps 6 or less which seems fair enough, he has 7-K.
If player two keeps an 8 when player one keeps his..

Then player two will...
Win 4, draw 3, lose 20 against the possible 27 cards player 1 can have Only about 20% win rate.

Won't player one do better if he swaps an 8 ?

Am I missing something here :-)

A big part of this game is that player two will know what player one swapped as soon as it happens. The fun part is when player 1 swaps a card that is actually higher than what is returned, and player two doesn't even have to hit. Conversely, the dealer knows if they swapped a higher card, they must hit to win. We don't play pushes in our poker game: low ties all must pay the pot. That's why when we play it, the deal rotates every hand. Huge advantage to player 2 (or dealer).

This is also why it's wise to hold 7 or better. Player 2 will not know whether player 1 has a marginal or good card, very valuable to player 1 that they don't know. Which is why player 1 should keep marginal cards luke 8 or 7. Player 2 has roughly 50% chance of having lower, is more likely to hit a marginal card if player 1 stands, and has a roughly 50% chance of hurting themselves if they do hit a middle card.

With more players than head to head, some will even stand on 5 or better, sometimes even 4, especially if previous players swapped and the 2nd one stood on what was a swappable card.

When we built this into a casino game (One for the Money, Ultimate Casino War, or Poker War, depending on how the casino wants to offer it), SHFL converted the Ace to high after they bought it. With ace high, the math says swap 2-7, stand 8-10, raise J-A. Lower those values by 1 each when playing ace low; they're still correct for the family poker game.

Fwiw, I love how Crown Melbourne is offering our game as Poker War. Take a look here if you're interested.

I don't understand that calculation
If P1 sticks he has 7-K
If P2 sticks with an 8... he will
W D L
4 3 20
15% 11% 74%

If P2 swaps his 8 he has to beat 7 - K

card # W % D % L %
7 4 23 46 3 6 24 48
8 3 20 40 2 4 28 56
9 4 16 32 3 6 32 64
T 4 12 24 3 6 36 72
J 4 8 16 3 6 40 80
Q 4 4 8 3 6 44 88
K 4 0 0 3 6 480 96

Sorting all that with weighted averages etc. I get
W D L
23% 6% 73%

Which beats sticking

Quote: Spartan62

If P2 swaps his 8 he has to beat 7 - K

card	#	Wins	W%	Ties	Tie %	Losses	Loss %
7	4	23	46	3	6	24	48
8	3	20	40	2	4	28	56
9	4	16	32	3	6	32	64
10	4	12	24	3	6	36	72
J	4	8	16	3	6	40	80
Q	4	4	8	3	6	44	88
K	4	0	0	3	6	48	96

Sorting all that with weighted averages etc. I get
W D L
23% 6% 73%

Which beats sticking

I am assuming that the rules say that the second player can't swap cards if the top card in the deck is a King. It looks like you don't use that rule.

If P1 has a King, P2 cannot tie (er, draw) by swapping as if the top card is a King, P2 keeps his 8.
Treat all Kings in the deck as 8s, as P2 will have to keep his 8 in that case, and calculate your numbers again.

Interesting rule, TDG! We never used that one. I see the rationale for it, though. Since with 2 players, p1 wouldn't expose his king until p2 had made his decision (in our version; perhaps in some rules, a player has to expose a king in turn to play it?), I wonder how many would hit the 8, though.

But, yeah, seen many dealers saved by drawing the king, and being able to play it.

Kings are turned up immediately regardless of the number of players. If heads up and the non-dealer has a king, the dealer can draw for a tie. We've played this way for 30 years since I learned the game.

Quote: Ibeatyouraces

Kings are turned up immediately regardless of the number of players. If heads up and the non-dealer has a king, the dealer can draw for a tie. We've played this way for 30 years since I learned the game.

In our friendly game, kings are only exposed if someone tries to swap for one. If unforced, the king stays hidden until showdown. Different regions, different rules, I guess.

Quote: beachbumbabs

In our friendly game, kings are only exposed if someone tries to swap for one. If unforced, the king stays hidden until showdown. Different regions, different rules, I guess.

The whole point of exposing it immediately is the blocker effect.