Principle of restricted choice

The principle of restricted choice is a guideline used in card games such as contract bridge to intuit hidden information. It may be stated as "The play of a card which may have been selected as a choice of equal plays increases the chance that the player started with a holding in which his choice was restricted."^[1] Crucially, it helps play "in situations which used to be thought of as guesswork."

For example, South leads a low spade, West plays a low one, North plays the queen, East wins with the king. The ace and king are equivalent cards; East's play of the king decreases the probability East holds the ace – and increases the probability West holds the ace. The principle helps other players infer the locations of unobserved equivalent cards such as that spade ace after observing the king. The increase or decrease in probability is an example of Bayesian updating as evidence accumulates and particular applications of restricted choice are similar to the Monty Hall problem.

In many of those situations the rule derived from the principle is to play for split honors. After observing one equivalent card, that is, one should continue play as if two equivalents were split between the opposing players, so that there was no choice about which one to play. Whoever played the first one doesn't have the other one.

When the number of equivalent cards is greater than two, the principle is complicated because their equivalence may not be manifest. When one partner holds ♣Q and ♣10, say, and the other holds ♣J, it is usually true that those three cards are equivalent but the one who holds two of them does not know it. Restricted choice is always introduced in terms of two touching cards – consecutive ranks in the same suit, such as ♥QJ or ♦KQ – where equivalence is manifest.

If there is no reason to prefer a specific card (for example to signal to partner), a player holding two or more equivalent cards should sometimes randomize their order of play (see the note on Nash equilibrium). The probability calculations in coverage of restricted choice often take uniform randomization for granted but that is problematic.

The principle of restricted choice even applies to an opponent's choice of an opening lead from equivalent suits. See Kelsey & Glauert (1980).

^ Frey, Richard L.; Truscott, Alan F., eds. (1964). The Official Encyclopedia of Bridge (1st ed.). New York: Crown Publishers. p. 457. LCCN 64023817. All subsequent editions of the Encyclopedia (up to the 7th in 2011) retain this definition. The original article was written by Jeff Rubens as is indicated by his initials at the end of the article in the 1st edition; the entire article remains the same through all seven editions of the Encyclopedia.

[1] Frey, Richard L.; Truscott, Alan F., eds. (1964). The Official Encyclopedia of Bridge (1st ed.). New York: Crown Publishers. p. 457. LCCN 64023817. All subsequent editions of the Encyclopedia (up to the 7th in 2011) retain this definition. The original article was written by Jeff Rubens as is indicated by his initials at the end of the article in the 1st edition; the entire article remains the same through all seven editions of the Encyclopedia.

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