## Questions we can say in words but cannot translate into precise mathematical questions

This is the style of "paradox" which I find most interesting. Here are three examples.

### 1. Chance vs skill in games

To say that there is no element of chance in chess is to take a very blinkered view of probability. There is a chance that your opponent will make a mistake, and a chance that you make a move which (unbeknownst to you) is brilliant. Having said that, compare a game of chess with a 2-person card game played for the same duration. Common sense says that, if we compare the skill/luck elements of the two games, then because of the randomness of card dealing there will be a greater element of luck affecting the outcome of the card game than the chess game. In other words we imagine one could compare games in some way like "chess is 90% skill and 10% luck; the card game is 50% skill and 50% luck".

The paradox is that one cannot quantitatively compare different games in this skill/luck way.

The point is that what we can estimate, from repeated games between the same players, is win probabilities (for simplicity I ignore draws). For instance suppose we find
player A will beat player B at chess with probability 75%
player C will beat player D at the card game with probability 60%.

Now suppose there were some absolute measure of "skill difference between players" that could be used in different games. The data above is compatible with two different scenarios:

• The absolute skill difference between A and B is the same as between C and D, but there is a greater element of luck in the card game than in chess.
• There is the same element of luck in the card game as in chess, but the absolute skill difference between A and B is greater than that between C and D.
And there is no way to disentangle these hypothetical notions of "absolute skill difference" and "extent of luck".

### 2. Making a sport exciting to spectators

What makes a sport interesting or exciting to spectators? Obviously there are many aspects to this. As one aspect, if a particular match is remembered as exciting, it will typically involve the result being unclear until mear the end. Can we design rules so that this first desideratum often happens?

The short answer is "yes", illustrated by the television show Jeopardy! Here the points awarded are doubled in the second half, and there is a final round where you can bet all your money. Such obvious manipulation of rules would not be acceptable in a professional sport. In that longer duration match setting, we want (as a second desiderata) the result to typically be influenced more than slightly by the first half scores; otherwise spectators will wonder why they should bother watching the first half.

To say the issue mathematically, suppose that initially the home team is assessed as having 50% chance to win. This probability will fluctuate throughout the game, ending at 0 or 1, and must be a martingale. Consider this probability at half time. It will vary from match to match. The first desideratum is that it not typically be close to 0 or 1, whereas the second desideratum is that it not typically be close to 1/2. Now as a non-obvious theorem, given any martingale with the properties above one can in principle design a game where the winning probability fluctuates as the given martingale.

So the problem is what specific martingale should we choose, to maximize excitement? Given a quantitative definition of "excitement" we can simply choose the maximizing martingale, so the problem reduces to making a definition of "excitement". And the "paradox" here is that there does not seem to be a natural definition. Here (more technical) is my best attempt.

### 3. The right mathematical version of the efficient market hypothesis

(Note: this is a more technical topic.)

In several areas of mathematics, such as computational complexity and cryptography, there is a meaningful and important distinction between "there exists ....." and "it is practical to find ........". This distinction is not currently made within mathematical probability, but here is a setting where it would be interesting to do so.

Consider a stochastic process $$X = (X_t, 0 \le t \le 1)$$. Theory around the optional stopping theorem states (in contrapositive form, and ignoring technical conditions)

• $$X$$ is not a martingale if and only if there exists a stopping time $$T$$ such that $$EX_T \neq EX_0$$.
For stock prices $$X$$, a "system" for making profit in the short term is essentially a stopping time as above. The essence of the (weak-form) efficient market hypothesis (EMH) is that there cannot be such "systems", because people would have discovered them and the effect of implementing them would alter prices until the advantage disappeared. This is the reason that short-term prices are modeled mathematically as martingales. But this argument really should deal with "systems that are practical to find". So to model prices we should really use some notion of "martingale for practical purposes" (MPP) defined by
• $$X$$ is not a MPP if and only if it is practical to find a stopping time $$T$$ such that $$EX_T \neq EX_0$$.
The problem is that no-one knows how to make sense of this, that is how to define "it is practical to find" in this context.

### Footnotes

1. You might argue that I am just considering game results, whereas one could instead look inside the game to identify more specific skill/chance events. That might enable a comparison between two different matches in the same sport, but does not solve the issue of comparing different sports.
2. A few attempts at formalizing "excitement" have been made, but none seem satisfactory to me. I learned the "non-obvious theorem" from Mykhaylo Shkolnikov. In a different model context, and roughly in accord with data, the half-time distribution (of home team win probability) is uniform on (0,1) -- see this paper.