# Probability intuition: Value or Ratio?

While doing some unrelated work, I ran into a dilemma on probability, which can be stated as follows.

In a game of chess, you can either lose ($$0$$ points), draw ($$\frac12$$ points), or win ($$1$$ points). At some specific moment, you can make two moves, $$A$$ or $$B$$. You know, for both the moves, the probabilities of winning, drawing, or losing. $$\begin{array}{c|c|c|c}&Win&Draw&Loss\\\hline A&0.2&0.6&0.2\\B&0.3&0.1&0.6\end{array}$$ Which move should you make?

Here's where I get a bit confused. $$B$$ has a higher win percentage than $$A$$, so I should play $$B$$. But in $$A$$, I have equal chances of winning or losing, while in $$B$$, I have twice the chance of losing as winning.

The expected point value of $$A$$ is $$0.5$$, and $$B$$ is $$0.35$$. However, I am playing only one game, so I should play $$B$$ because it has a higher winning percentage.

Which move should I play?

• It depends a lot on what your goal is. Are you trying to get the highest point score, or do you only care about winning? Jan 1 '20 at 20:05
• I would like to win, and if that is not possible, draw, and avoid losing. Jan 1 '20 at 20:09
• Winning is possible in both scenarios. The one that you chose depends on your priorities. If winning is an absolute priority then you choose B. If the point value is the absolute priority you choose A. If not losing is the absolute priotity then you choose A. Jan 1 '20 at 20:11
• The math can't tell you what priorities to have, does that make sense? Once you have clearly identified them, then you can use the probabilities to make a rational decision. Jan 1 '20 at 20:12
• Given that the task gave those points, I guess the intended quantity to optimize was the expected number of points. I doubt real life chess players think in probabilities anyway. Jan 1 '20 at 20:44

As already stated in the comments, the objective function needs to be specified as part of the problem statement and can't be deduced mathematically.

Near the beginning of a round-robin tournament, the expected utility a player derives from a game result can be approximated by the expected number of points. In this case, you should play $$A$$, as the expected number of points is $$\frac12$$, whereas it is lower for $$B$$.

Near the end of a round-robin tournament, especially in the last round, a player typically knows which result she needs in order to win the tournament. For instance, you may need to win the game in order to win the tournament, or you may know that you’re sure to win the tournament if you can secure a draw. In this case, if you need to win, you should play move $$B$$, since it has the higher probability of winning, whereas if you want to secure a draw, you should play $$A$$, since it has the lower probability of losing.

As others have said, it depends. Decisions require probabilities, value of outcomes, risk-aversion, etc.

• If you get your head cut off when not winning, play B.

• If you get your head chopped off when loosing, play A.

• If you get the number of points in dollars you can use expected values. A is then worth $$.2+.6/2=.5$$ and B is worth $$.3+.1/2=.35$$, hence play A.

• If you get the number of points in billions of dollars you may want to play it safe and play A.

ps. I am assuming you are not infinitely rich and want to keep your neck intact. :)