Bene Factum

I’m a big fan of poker, and of bluffing games in general. If you’re keeping up to date with me, you know that my next release, which will be on shelves soon, is Cash or Crash, a card-based bluffing game that is an elaboration on the basic concept of the classic Liar’s Dice.

This morning, I had an idea for another bluffing game, but one so simple that I could never market it in its basic form, and indeed would be surprised if I turned out to be the first to have come up with it. I haven’t tested it yet, but it’s one of these games that’s so simple as to be self-evident; it’s probably susceptible to mathematical analysis, but if humans have a hard time playing Rock-Paper-Scissors randomly, I doubt that a computer generated table of optimal move-selection probabilities would help anyone become unexploitable in this game.

I am going to start by explaining the game as originally envisioned, as a two-player game. Additional multi-player rules will be given afterwards – both a simple version, and a more elaborate one. The simple version is much the same as the two-player game, but suffers from the problem of being highly dependent on seating order, and would be too easily abused by colluding players sitting next to one another. The more elaborate version would be much more appropriate for cash play, though of course I cannot recommend gambling if you live in a place where gambling is illegal, or are under the legal age.

Here it is:

Mentalist Poker

1. Each player starts with 20 chips or counters of some sort, a screen to hide them, and a “betting line,” perhaps a piece of string or a pencil, placed horizontally in front of him (but behind the screen). Chips placed in front of the line are being bet, while those behind are not.
2. Determine at random which player starts with the First Bettor button.
3. Each round begins with both players making an ante of one chip.
4. The First Bettor secretly decides on a bet, which can be anywhere between zero chips and all of his chips. He places these in front of the line, being sure to disguise his motions.
5. The opponent attempts to guess how many chips he has bet, stating verbally “Zero,” or “Five,” or whatever number he likes.
6. The bettor’s screen is lifted and his bet revealed. If the opponent’s guess was correct, the opponent immediately wins the pot and the bettor’s bet.
7. If the opponent’s guess was incorrect, the bet is added to the pot, the screen is replaced, and now it is the opponent’s turn to make a bet, and the First Bettor’s turn to guess.
8. If it is a player’s turn to bet and he has no chips left, the opponent is assumed to guess Zero, and automatically wins the pot and the game.
9. Play continues back and forth until the pot is won. Assuming both players have at least one chip left, the First Bettor button is passed to the opponent, both players ante again, and a new round is begun.

And that’s it! A player would obviously like to bet Zero as much as possible to avoid adding chips to the pot, but cannot do this all the time, or it is too easy for his opponent to collect the antes. Betting One or Two instead is likely to buy the player a chance to take a guess of his own, but as the pot grows, so too does the range of bets the opponent might profitably make. On the other hand, as players approach a situation where all their chips are in the pot, the incentive to bet low begins to grow once more, as a player who commits all his chips now has only one more chance to guess correctly and win, or else lose everything. For this reason, a player who gets a chip advantage also gains a strategic advantage in subsequent large pots, thus fighting for the lead may be worth larger gambles than poker-style “EV” calculations might indicate.

Simple Multiplayer

The rules are exactly the same as for two-player, except that it is important that the Bettor be betting for the right to make the next guess. Thus, the Guesser is always the player to the right of the current Bettor, while the next Bettor will be the player to the left. The First Bettor button likewise passes left after each round. In other words, the betting passes around clockwise as in most games, but the Guesser is always the last person to have bet (or the player who was First Bettor in the last round).

(The reason this is vulnerable to collusion should be obvious – if I am sitting to your left, we agree beforehand that I will bet in a systematic way, allowing you to “guess” correctly most or all of the time and win the pot, presumably in return for a cut of the winnings, or for you allowing me to win when our positions are reversed. It’s equally bad for everyone else if I’m not colluding with you, but am simply a terrible player who e.g. bets Zero way too often.)

Advanced Multiplayer

1. Antes are posted, and a secret bet selected as usual. No “First Bettor” button is needed, as play will pass continuously to the left.
2. Larger chip stacks are required, as least equal to 10x the number of players. I.e. in a four-player game, each player should start with at least 40 chips. Multiple denominations are likely required, with players able to make change freely.
3. The bet is capped at the current size of the pot. For instance, in a five-player game, with only the starting antes, the maximum bet would be Five.
4. Instead of there being a single Guesser, everyone except the Bettor makes a guess, starting with the player to the Bettor’s right (i.e. the previous Bettor or the First Bettor of the previous round) and proceeding counter-clockwise.
5. If anyone guesses correctly, the Bettor says so immediately and lifts his screen, and that Guesser wins the bet and the entire pot.
6. If everyone has made a guess and no one has guessed correctly, the Bettor lifts his screen to reveal his actual bet. Unlike the standard version, the Bettor keeps his bet, and then claims as many chips from the pot as he had bet.
7. Regardless of the outcome, everyone now antes one additional chip, and the next player to the left becomes the Bettor.

Example: In a five-player game, there are 5 chips in the pot to start. If I am Bettor, I can make a bet between 0 and 5, i.e. 6 possible choices. The other players will have a total of 4 guesses. If I bet 3 and the guesses were 2, 1, 5 and 0, no one has guessed my bet. Therefore, I can claim 3 of the 5 chips from the pot, leaving 2. Everyone, including myself, now antes one more chip, making a pot of 7, and the player to my left must make a bet. As an added bonus, I will now get the first opportunity to guess.

The strategy here is considerably different than the basic game, in that the risk-reward calculations for the Bettor are different. With multiple guesses instead of just one, there is a much larger chance of having my bet guessed. Thus, there is a temptation to surrender the pot by betting Zero a lot, and simply trying to win pots when I’m to the right of the First Bettor. If my bet went into the pot on top of this, I’d have an even further disincentive to bet aggressively, thus the game would be pretty boring. That’s why, in this multiple-guesses version of the game, a successful bet is taken out of the pot, rather than added to it. The pot size will also fluctuate constantly, rather than growing steadily until someone takes it down. The stakes (and thus available options) can grow quite large indeed if multiple players in a row make small bets that are not guessed.