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).

How to solve a math problem, the engineering way:

1. Dimly recall that you studied it somewhere.


2. Dig up the likeliest book.


3. Confirm that it describes the solution.


4. Balk at the math involved.


5. Download a tool that solves the problem in a second.

To recap: while playing Lord of the Rings: the Confrontation, we have a battle we are winning 3 to 2 (before cards) and both players have the cards 1-5.

In running the calculation, I had to assign a payout matrix.

	1	2	3	4	5
1	10,-10	-2,2	-9,9	-7,7	-5,5
2	8,-8	10,-10	-2,2	-9,9	-7,7
3	6,-6	8,-8	10,-10	-2,2	-9,9
4	4,-4	6,-6	8,-8	10,-10	-2,2
5	2,-2	4,-4	6,-6	8,-8	10,-10

How to read that: Left side are our cards choices, top side is our opponents choice. If we both play the same number, then the result is (10,-10). That's the best possible result for us, since we win the battle and don't lose any ground. As a frame of reference, I valued winning a piece at 10 points. Then I called each pip two points. If we tie (my opponent plays a card one higher) then both pieces die. I'm going to call that a marginal victory for my opponent (reasonable if playing Sauron), but mitigated because he used a better card. Similarly, if my opponent wins by playing a 3 to my 1, I did't call that the full ten points, but the loss is more than six points (which I should do if using 10 points for winning and 2 point per pip difference).

Optimal play according to Gambit:

• Play the 5 and guarantee the win 62.4% of the time.


• Play the 1 (and try to win cheaply) 14.3% of the time.


• Play the 3 (and prevent the countershot) 11.0% of the time.


• Play the 4 (risking the draw, crushing a middle card) 7.2% of the time.


• Play the 2 (why not?) 5.1% of the time.

I'm glad that I put the first three choices in the correct order, but I expected a more even distribution. Discounting the 4 & 2 (as I did), I took the five 54% of the time (rounded to half). But even with two more choices, theory says take the win nearly 2/3rds of the time. This is much sharper than a power distribution.

You can quibble with the assumptions. Is one card pip really worth 2 points (given that a piece is worth 10)? I tried it with 1 point and playing '5' shot up to 80% (with my opponent conceding with a '1' 80% of the time). Intuitively that makes sense: as pieces become more valuable you take the win more often. Conversely, if I increase the value of the cards in relation to the pieces, playing the '5' becomes a weaker choice.

One ignored effect is that by playing 5:1, I may lose the ability to force a win in the next battle. Let's model that by changing the (5,1) cell to a payout of (1,-1), while leaving the rest of the model untouched. The result leaves the 5 at 60%. Amazingly, the second choice shoots from 14.3% to 18.6%, absorbing a few points from the worst result (the '2', which falls to 2%). But the rank ordering doesn't change.

Don't read too much into the 'optimal' result. [Optimal doesn't mean best, which is to know the card my opponent will play and react accordingly. Optimal means you can't improve on this against a perfect opponent. Opponents are rarely perfect.] In addition, this payout matrix is artificial. The value of a piece changes according to the game situation.

The dizzying insight: don't overthink simple situations. Even rating cards highly, pick the obvious play more than half the time, closer to 2/3rds. But remember the important caveat: this assumes a sure win. Real games rarely cooperate...