The Tao of Gaming

Playing Lord of the Rings: The Confrontation, simultaneous decision problems loom in my mind. Consider a simple case: I have a '3' character, my opponent has a '2.' Each of us must play a card 1-5. High total wins. I can guarantee the win by playing my five. My opponent, knowing that, can play the '1'. Since you lose the cards until you've played them all, I win but now have a much worse hand for the next battle. Knowing this, I could play a '1' to win the battle but keep card parity. So my opponent could play a '3'.

Game theory provides an answer (assuming we can decide on a payout matrix); but geeky gamers know game theory. Unfortunately I can't do it in my head except for simple cases, and 5x5 matrices are "non-trivial". So it comes down to psychology, reading the opponent, and evaluating the board position ["Can I afford to lose this battle if I guess wrong?", "Card value isn't important now" and other questions outside of the scope of our little thought experiment...]. In short, I guess.

Against novice gamers, my instincts usually guide me true. (In fact, I taught LotR: the Confrontation to a new gamer and my win embarrassed me). Against experienced gamers, I constantly second-guess myself and go one level too deep or one level too shallow in my thinking, and invariably wind up with a fatal dose of Iocane powder.

So, I'm wondering if there is a rule of thumb that may help. Perhaps my decision should follow a Power Law. So, If I have "N" levels if "If he knows that I know" that means:

1. Pick the obvious move some percantage X, in our example, play the 5 and win.


2. Pick the move that responds to the expected counter percantage X/2 ("Play the 1 and win cheaply, as your opponent will play the 1")


3. Pick the move that responds to the expected counter-counter X/3 percent ("Play the 3 to counter the 3"


4. Keep going throught the choices...

So if I had three levels of thinking, I have x+x/2+x/3=100%, so 11x=600%, so x=54%. The final answer: take the obvious move 1/2 of the time, the counter 1/4 of the time, and the counter-counter 1/6, (and the 4th level 1/12, etc). Perhaps I'll work this case out analytically later to see what it says. I'm sure the results will surprise me. (Say, does anyone know of a good site that will do this calculation for me?)

This psychological/analytic choice appears in many guises. I suppose the most common form is a blind ("In the fist") auction, or a "Once-around" where you bid before all/most of the other players. Adel Verpflichtect took this idea (and scant else) and turned it into a classic. Interestingly, I don't consider Poker to fall into this category. The cards will win or lose on their own, and I'm merely trying to maximize my wins and minimize my losses (by bluffing, folding losing hands, betting for value, etc).