Hi guys this may be long winded post here about some game but really appreciate it if anyone could help me out. It's an interested spot on debate between me and some friends of mine.

The basics card of the card game are simple, each player is dealt 5 cards. Each player lays one card per round and whoever has the strongest card value at the end of each round wins what is known as a trick and it equates to 5 points. The goal of the game is to get to 25 points to win that entire game. It can take multiple rounds to get to 25 and there are new deals after everyone has all 5 cards played.

Trumps play a vital role in this game, after the 5 cards are dealt, a card is turned up from the top of the deck and that suit is the trump for the round giving them preferential strength over every other suit. The trump turned over can also be "robbed" or replaced with the worst card in the persons hand if they are in possession of the Ace of that suit.

The card rankings are as followings-
5 of trumps is the strongest card.
Jack of trumps is next.
Then the ace of hearts is always powerful regardless of suit.
Then the ace of trumps.
Then it ranks from the rest with the K being the strongest right down to the 2 being the weakest trump but beating any other card in the deck.

So this particular hand occurs.
The scores are:
Player A 20
Player B 20
Player C 20
Player D 15

So only player D can win the first trick without the game being over and a winner for that round. But it will only every go a maximum of two rounds.

So Player A lays the 6 of trumps out of his five cards. This will be his strongest card given the action. He will have 4 other cards which may contain weaker trumps than the 6 but no guarantee.

Player B has two trumps in his hand the 3 of trumps and the K of trumps of which he has to lay one or the other. He has three other none trumps in his hand which will play no factor in this hand but for probability we can take those out of which is the correct lay here probability wise if it can be worked out?

If he lays the 3, he is relying solely on Player D winning this trick. If player D has nothing it is game over with either player A or C winning depending on their lay. So for example if both player C and D have nothing Player A wins the trick with the 6 and therefore game. Player C can then win if he lays the 7 or better and player D can't beat it. So player D must win for us to lay our 3 and still have the K.

If he lays the K of trumps. He needs to beat the two players after him which means he wins provided neither has the 5 of trumps, Jacks of trumps, Ace of hearts or Ace of trumps.

There is also the factor of of Player C beating his trump but Player D then beating him and thus allowing his three of trumps to be a possibly factor.

Should he mathematically, all things considered lay the King here and hope to beat the two players behind him? Or should be lay the 3 and hope that player D can beat both players A and C.

Any questions feel free to ask as it might seem a tad complicated but any help you can give me would be greatly appreciated. Thanks.

I have only skimmed the post, but it may be important to better understand what would subsequently happen if Player D wins the first trick.

For example, who plays the first card on the second trick? Do the rules of the game require players to "follow suit"? That is, if Player D wins the first trick and plays the Qh on the second trick, do other players have to play a heart if they have one or can they play a trump card whenever they want?

If so, the three other cards in Player B's hand may be important (voids, aces, etc.).

Also, relatedly, do you have to play a trump if a trump is led? This may come into play depending upon what Player C has.

I hope these questions make sense.

Hey. Yeah the questions make perfect sense.

If player D wins the trick, he will be first to act on the next trick and the action will end at player C. You must follow suit, but as it's next trick wins if player D wins the first everyone will be playing their strongest value card given it's the deciding trick.

If someone played the Queen of hearts and diamonds were trumps you must either follow suit in hearts or lay a diamond(obviously if you don't have either nothing you can do and just throw in your worst card).

If a trump is played you can only renege(hold back) the ace of hearts from regular trump. However. if the Jack of trumps is played you must play the ace of hearts if it's your only trump. Same situation from the 5 of trumps. You can renege the Ace of hearts, Jack and 5 of trumps provided a superior card hasn't been pulled.

Like I said all questions welcome in determining if it's the best play to play the K here.

A requirement to "follow suit" means that if a heart is led (assume diamonds are trump) and if you have a heart and a diamond, you must play the heart (following suit).

Is this the rule in this game or could a player play his trump in this case?

Nope you can play the heart or the diamond. You could not play a club or spade if you have a heart in your hand. However, you are under no obligation to play the trump.

In general, I wouldn't be so quick to rule out Player A having higher trumps than the 6. Suppose he was dealt the Qt and 6t, say. Then leading out the 6t may well be the better play than leading out with the Qt. Of course, he would be trying to "force out" higher trumps than his Queen on the first trick so that his Qt can win the second trick (hoping that player D wins the first trick with a high trump).

Anyway, you advise that you want the optimal decision for Player B choosing between playing his Kt or 3t on the first trick assuming that Player A has no trumps higher than the 6, so people should follow that assumption.

One "challenge" with these types of problems is that the optimal solution could well depend upon how the other opponents will play their cards. For example, if Player C has both the Ah and Qt, I suppose it is possible that he would play the Ah if you play the Kt and play the Qt if you play the 3t. Having never played the game, I have little "feel" for the best strategies.

For ease of solving, it would probably be best to assume that Players C and D will play their highest trump on the first trick (if they can take the trick) and their lowest trump if they cannot (if they have more than one trump).

I assume that, with these assumptions, determining whether Player B should play the 3t or Kt on the first trick will be feasible.

I may look into it this evening if nobody posts anything before then.

P.S. I think we also need to know what was the turned up card and who, if anybody, "robbed" trump on the deal.

Keeping commonly used strategy and player tendencies I would safely assume the 6 was the best trump Player A had. For this lets just assume that's the case.

Your assumption is correct but at the moment we can assume both player C and player D's hands are random. Is it possibly to work it out through maths which is the better play? Two players need to not have the top 4 cards. Then we need player D to win if we lay the 3. First we need player A's 6 beaten by player D. Then we need player D to beat player C if he improves upon the 6.

The turned card was the 2 of trumps. In this game, you don't need to "rob" until the decision gets to you so we aren't privy to the information on whether C or D have the ace. All we know is player A cannot have the Ace and us(player B) doesn't have it either.

Any help would be greatly appreciated. Thank you.

Last question (I hope): trump was not hearts, right? So there are essentially 14 trump cards.

Diamonds were trumps in this particular hand.

Player A shouldn't be leading a singleton trump in this hand below the At. The rest of the hand is basically absurdly tedious to enumerate and figure out the game theory aspects of, and it's close enough that you need to do a bunch of work.

Playing the 3t is shooting for a D-wins-then-we-win parlay. For it to be the right play, P(D wins the first trick) × P(Kt wins the next trick | D wins the first trick) has to be greater than P(Kt wins the first trick).

Let's assume you're right that A has no higher trumps than the 6t. That means he has 4 unknown cards that don't beat Kt. Using that information, the 6t, the turned 2t, our 3t and Kt, and our 3 other cards that don't beat Kt, the probability that neither player C nor player D can beat Kt is:

P(Kt will win the first trick) =
(37/41)(36/40)(35/39)(34/38)(33/37)(32/36)(31/35)(30/34)(29/33)(28/32) ≈ 31.1%

We'll assume that your chance of winning after Kt busts is negligible. This will only be worth revisiting if P(D wins the first trick) × P(Kt wins the next trick | D wins the first trick) > 31.1%, but it's close.

If we play the 3t, not only must player D be able to beat 6t and whatever C plays, but our Kt must also be playable on the next trick (so your other suits matter a lot here) and must not be beaten by anyone.

The math is more complicated, and I don't have the time to hash it out right now, but I strongly doubt that it's greater than 31.1%.

Tentative decision: Play the Kt on the first trick.

By the way, the 3 other cards in our hand do matter even if they're not trumps.

They don't make a huge difference in figuring out if Kt will win the first trick, but they're significant toward figuring out our chances on the second trick if we dump 3t in the first.

You can trump even if you could follow suit.

I fiddled around with this problem last night. To make the problem most amenable to analytical methods, I have assumed that Player A's only trump is the 6d and he will lead it on the first trick. I have assumed that Player C will play his highest trump on the first trick if he has a higher trump than the highest trump card played on the trick so far (otherwise he will play his lowest trump). If Player D can take the first trick, he will play his lowest trump that takes the trick (or otherwise his lowest trump). If there is a second trick, of course everyone will play their remaining highest trump card. I have not yet incorporated the switching (robbing) that the rules allow if either Player C or D has the Ace of diamonds. I doubt whether that would have a significant impact on the results. Any of these assumptions can be modified but I wanted to start simple.

Anyway, my method was to tabulate how many trumps could be dealt collectively to Players C and D, and then tabulate the frequency of Player B winning the game under both strategies (playing the Kd or 3d on the first trick) in that scenario.

If anyone is interested in the details I could provide them, but the following are the high-level results.

Player B will win 39.9% of the time by playing the King of trumps on the first trick and will win 35.4% of the time by playing the Three of trumps on the first trick. So playing the King is the preferred strategy.

I am in the process of coding up a simulation that can serve to confirm/deny these analytical results. One problem I fear in advance is that my simulation will borrow heavily on the code I have written to tabulate the winning frequencies. So I am not sure the simulation results will be sufficiently independent from my analytical results.

When I have some results of the simulation I will post them.

I ran a simulation of 100,000 deals and tallied how often each strategy would win for Player B. Playing the King of trumps on the first trick wins 40.710% of the time and playing the Three of trumps on the first trick wins 32.938% of the time.

The first winning percentage is close to the pct I found analytically (39.9%) but the second winning pct is only fairly close to the pct I found analytically (35.4%).

Not sure if this simulation completely supports the analytical results or not.

Wow. I'm a little surprised that the analytical probability for 3t winning the second trick, given than Kt loses the first, is 8.8 percentage points higher than P(Kt wins the first trick). Not so surprised now that I think about it, but I wasn't expecting it to be that much last night.

Do your analytical calculations account for the cases when C beats our Kt and then D beats C? That would remove one more card that beats 3t for the second trick, perhaps accounting for the slightly higher probability found by the simulator.

I have essentially dealt out every possible pair of hands for Players C and D and then had each player play according to the strategy I outlined above for two tricks (if D wins the first trick).

So yes, any possibility like the one you have mentioned is fully taken into account.

Wait, what exactly do you mean by "analytically," then? That's what I was asking about. I thought you meant that you calculated 39.9% using compound probabilities rather than from a simulator.

I wrote a program that dealt out every possible combination of 0-5 trump cards to each of Player C and D. The program played out every game until there was a winner. It kept track of how many games Player B won using the twin strategies. These win pcts were then weighted by the likelihood of getting that deal.

After writing that program I then wrote a standalone simulator program to deal 5-card hands out of a standard 52-card deck to both Player C and D (the hands of Player A and B were assumed fixed). The simulator then, as above, played out the tricks and kept track of who won the game.

If there are no mistakes anywhere, the two approaches would give similar results.

Man, I'm pretty jealous of your programming ability. I have a grip on most of the basic math, but being able to simulate a game at will would be really helpful sometimes.

Thanks for the comment but my programming ability is actually quite limited. I have admitted before that most of my current programming endeavors are connected to trying to teach my daughter programming concepts and methods. Things like Yahtzee and Poker are things that she understands and enjoys, so they make fertile ground for simple computer programs.

You might be surprised how little programming knowledge is needed to write most sims. If you can write loops and "if" statements, you can write a sim. Probably in less than a day you can google about the language of your choice and teach yourself how to write, say, a sim to check a Roulette probability or test a system. Hell, if you have a programmable calculator you can write one on that and you wouldn't even need the loop commands, you could use Labels and Goto's.

Yes, I am definitely self-taught and learned how to program because the early PC's came with a version of Basic (I think it was called GW-Basic). There are many things that spurred the PC revolution, but one of the aspects was the capability of writing your own computer programs.

Today the tools, guides, and choices are more prevalent, so it is much easier for anyone interested to learn how to program.

Hi whosnext, thanks for working out this problem for us, I was another player involved in this game. One more factor if you can run it is I would safely assume that player C using player tendencies would lay his lowest trump better than the 6 if player B lays the 3. Would be grateful if you could work out the results using this information.

Under this scenario for how Player C would play, the probability of Player B winning by first playing the 3d lowers slightly to 30.7%.

Thus, even under this scenario, it is still better for Player B to play his Kd on the first trick (40%).