Schrodinger's Card

In 1935 Erwin Schrodinger devised a thought experiment whereby a cat was placed in a box with a vial of poison, a hammer, a radioactive source, and a Geiger counter. If the Geiger counter detects radioactive decay it lowers the hammer, breaking the vial and killing the cat. The Copenhagen interpretation of quantum mechanics treats the cat as being in both an alive and dead state, until such time as the box is opened and the fate of the cat is discovered. But what does this have to do with games?

Quite often when playing a game there will be some hidden information, a face down or as yet undelt card, some undefined counters, or even the result of the next dice roll. When we make our decisions we need to consider what these still to be determined events could be and the risk/reward in each case. For example, playing The Settlers of Catan I have nine cards in hand on my turn and have to make the choice between using some of these cards or keeping them to use later, the risk being that if I choose the latter a seven will be rolled and I'll lose half of them. How should I make my decision? Well I need to consider the probability of a seven being rolled and also the probability that if it isn't I will be able to make better use of my cards next turn.

Let us now switch games to cribbage. At the start of each hand a player must decide which two cards to place in the crib before the starter card is revealed. Because the starter is used at the end of the hand for scoring consideration to how our kept cards can combine with the starter is important. Since sixteen of the cards have a value of ten and only four a value of nine, keeping a five is more beneficial than keeping a six (having cards that add to fifteen scores two points). It is also worthwhile, as explained in my How to get Lucky post, to consider the chance of keeping a hand that is not currently high scoring but with the right starter card becomes high scoring. Some people find this kind of thinking difficult, as the obviously the top card is already determined, and also your opponent has six cards in their hand so none of these six card can be on top! However, probability doesn't understand that and we must treat the cards like Schrodinger's cat, being all cards at once until such time as they are observed, except for one small problem.

Our opponent is not random, but rather is making decisions all the time and based on these decisions we can get a hint about what cards are where. In 2008 a Nature article detailed scientists 'peeking' at the cat without causing it to become either alive or dead. When our opponent plays their first card we immediately gain some information and get a 'peek' at the cards in their hand, for example if our opponent leads a ten I would assume that they have a five, I don't know for certain so the card is still undefined but I get a hint. The reason is that, as I explained earlier, the starter card is most likely worth ten, also the most common card value in your hand is a ten which means that keeping fives is very common, meaning the chance of someone having a five as one of their four kept cards is closer to 2/5 than 2/7, so by leading a ten I assume that they have a five to play (scoring two points for doubles) in the event that I play my five.

This type of thinking also applies to poker, when a player folds rather than bids we gain information about what they don't have in their hand, and the probability that those cards will appear in the community cards increases, of cause poker also gives us the option of lying which is where knowing your opponents play styles and 'tells' comes in handy. I once used knowledge of my opponents play styles and 'peeking' to great effect in a game of 500. Due to time constraints it was the final hand, I had dealt and the player to my left opened the bidding at the eight level which my partner then matched, the player to my right then made a bid at the nine level. I still hadn't looked at my cards but I had played with these people a lot and knew they were not the types to overvalue a hand even if it was the final had of the day, so still without looking at my cars I bid open Misere, after some complaining about bidding blind and me wasting their time the bid was accepted and made, winning my partner and I the game. Afterwards I pointed out that because of the way they had bid I knew my hand had nothing in it and that was why I was comfortable making such a large bid without having seen it. Despite this explanation I still got a reputation for being a maverick, crazy player who despite making stupid bids would end up winning.