# Poker justice

I played poker last night. Texas hold ’em, apparently. The hand you bet against is the best five-card hand you can muster from two cards dealt to you and five that are dealt face-up on the table.

You’re dealt two cards each. Then there’s a round of betting. Then three cards are drawn, all face up on the table. Another round of betting. Card four on the table. Betting. Card five. And final betting.

£10 was the initial stake, with unlimited buy-ins before 10pm, after which, you bet until you won or you were eliminated.

Just before midnight, the nine players had been whittled down to two, of which I was lucky enough to be one. The £200-worth of chips (and supplementary hand-written 100-chip notes) were pretty evenly divided between the two of us, so instead of battling through the night we thought it wise to split the pot. £80 up on the night: not bad at all.

The reason for the post, however, is to question one specific feature of the game. If a player runs out of chips during betting, they can still stay in for that round, not being obliged to bet thereafter. If they win the round, then their winnings from each other player are capped at their own bet. So if Alberto has bet his last 25 chips, Balthazar 70 (before folding) and Cecilia 90 (still in), then Alberto winning means that they will take 25 from each of the other two players.

At first glance this seems quite fair. But the scenario has always troubled me, and I’ve just realised why. Let’s assume that the three players’ hands have equal chances of winning, and it is merely the playing styles of the players that are at odds.

If Balthazar and Cecilia get into a bidding war after Alberto has maxed out, Balthazar folding when Cecilia raises that bit too much to justify him continuing, then the increased investment by Balthazar and Cecilia has benefited Alberto, as he now only has to beat one hand rather than two. So Alberto has benefited from the additional investment of Cecilia, through it driving Balthazar out of the game.

Further, continued investment from one of the cash-rich players cannot result in them winning by default—they will always have to beat Alberto to win the round.

The trade-off is that Alberto can win fewer chips than Balthazar or Cecilia. But there is no trade-off for Balthazar or Cecilia.