The role of the house in Pai Kau betting -- beware!

On this page, I shall briefly describe the game of Pai Kau (or more precisely a simplified version of it that has the same basic structure). Later, we shall look at the way the house, at least in Macau casinos where I observed the game closely, handles the bets.

The game description: The game is too complicated to explain in detail (because of a few messy wild-card rules which affect the otherwise simple structure), so I shall explain a simpler (and hence slightly inaccurate) version. There is a deck of 32 tiles comprising 16 pairs numbered 1, 2, ..., 16. The tiles are randomly dealt to (up to) 8 players, a banker and (up to) 7 gamblers. Each player gets 4 tiles.

Each player must then (in secret) sort his hand into two groups of two -- a front duo and a back duo. The rules of the game specify a rank order for a group of two tiles. The best is the natural pair (16,16), then (15,15), ..., (1,1). After the natural pairs, two tiles (x,y) are ranked according to the size of (x+y) mod(10); the higher the sum mod(10) the higher the rank. If two duos have the same sum mod(10), the duo with the highest numbered tile is deemed better (so (16,2)>(5,3) because 16>5). If the tie is still unbroken, second-highest tiles in each duo are compared (so (13,11)>(13,1) since 11>1). There remains the possibility of a genuine tie if duos are exactly the same.

In sorting one's hand, one is constrained to make one's front duo of higher rank than one's back duo. So if a player is dealt (16,11,10,3) he has three choices of sorting:

Although there are many players, the game is effectively played as separate comparisons, banker's sorted duos versus each gambler's sorted duos. In each of these comparisons, the front duo of banker's hand is compared with the front duo of the gambler's; likewise the back duos are compared. If either player wins both the front and back comparisons, he wins the bet. If the success is shared, the game is a draw and the bet is annulled. If there is an exact tie in the comparison of duos, then the banker is deemed the winner of that comparison.

The betting protocol: You may marvel at the very low bias in favour of the banker, but before thinking that the Macau casinos are very generous you should know that the casino is not the banker. The banker is one of the players bold enough to take on the role when his turn (by rotation about every 100 deals) comes up. The casino deals the tiles and supervises all the money flows AND takes a commission of 4% on each winning bet. They also underwrite the banker to a limited extent and, as we shall see, cream off considerable profits when the banker wins a lot

The player who is Banker puts a sum of money, say B, in front of him before the tiles are dealt. Each of the 7 gamblers also place their bets.

The order of events is as follows. If you are a player, read carefully!

• The tiles, which are like big domino-tiles in appearance, are dealt. Each player sorts his tiles (secretly) into front and back duos. He then places them onto the table face down.
• The Banker's hand is then exposed by the croupier. ("Exposed" by slamming the banker's tiles on the table, face up. The gamblers either groan or yell in delight. The noise made by slamming the tiles is a vital ritual to the Chinese gamblers in Macau. Twice I saw tiles break!)
• The croupier then exposes the tiles of each gambler in turn, from 1 to 7. (This is done quietly!)
• Any gambler who has a losing hand, in comparison with the banker, has his money moved into the centre of the table. It is not moved into the Banker's pile of money at this stage.
• Gamblers who have drawn with the Banker are given their money back.
• There remains the issue of those who have won. In turn, they are paid out at even money (less 4% which the croupier takes). The payouts come initially from the central pool, but if this has no money, the payout comes from the Banker's amount B.

You can see the problem looming, particularly for a gambler with winning tiles sitting in seat 7. What if both the central pool and the Banker's amount B are depleted? Will he get paid?

Here the "house guarantee" is envoked. If gambler n wins and his bet x does not exceed an amount C (a sum of money that the house has announced for the table), then he will be fully paid (less the 4%), either in the regular way -- from the central pool or from the Banker's B -- or by the house as a last resort. If x > C and the Banker has insufficient funds, then a winning gambler receives C (less 4%) -- any shortfall from the banker being paid by the house.

One would think that the gamblers might keep their own bets within the threshold C if Banker's B is not very large. No, not always! Sometimes people bet without much regard for the magnitude of B or C. I saw one gambler who bet $HK30,000 on a deal in position 3 when the house guarantee C was only $HK3,000. The Banker drew poor tiles and his B of about $HK15,000 was cleaned out by players in positions 1 and 2. The man in position 3 had winning tiles but was paid only $3,000 (less the 4% commission). Yet, had he lost, he would have lost all of his $30,000.

Now we turn to what the Banker's profit is, if the central pool is still in surplus after all winning gamblers have been paid out.

• If there are enough funds in the central pool, the banker gets paid at even money less 4%. So he gets paid 96% of B.
• If there are not enough funds for this payout, he just gets 96% of the central pool.

Who takes the rest of the pool's surplus, you may ask? The house, of course! Is this fair compensation for their role in underwriting the Banker? I don't know, but it seemed from observation that the house was onto a "nice little earner" (as Arthur Daley would say) because of the irrationality of gamblers' betting. It seemed that the casino took more money via this mechanism (nett of any underwriting payouts) than by the 4% commissions, but I cannot be sure.

A few other games in Macau, for example the novices' game called "Marjong Pai Kau", have the same betting protocol.

Footnotes:

• A few years ago, a syndicate came to Macau and outsmarted the casino for a while before getting tossed out. When one of their number became Banker, he put forward a fairly small bank $B. In addition, he made bad sorting decisions purposely, thereby setting up losses for himself and gains for all his friends. On many deals, he lost all of his Bank and so the house had to pay his friends $C each. In nett terms, the syndicate gained from the casino. Nowadays, the croupiers will intervene if a Banker tries to make a poor sort of his tiles.
• Further detail about the game and its mathematical theory can be found on my linked page about block games.