## Sunday, September 15, 2013

### How to eat a meal

Many times my family and I face a dilemma when we go to a restaurant for a meal.  We would order a few dishes and when the food arrives we need to solve the following problem: there are several types of food on the table, each with a different preference in our mind. The question then is: which food should we eat first, which food should we eat second, etc.  My strategy is to eat my favorite food first.   My wife has a different philosophy; she saves the best for last.  I don't agree with her as I believe that your favorite food will not taste as good at the end of a meal, much more so than for food that you do not like as much anyway. Attempting to formulate this as a mathematical problem, we are essentially maximizing our enjoyment of the meal:
$$\max_S E(S)$$
where $F$ is the set of food items on the table, $S = (s_1,s_2,s_3,...)$ is the sequence of food to eat subject to the constraint that $s_i\in F$ and $E$ is your enjoyment of your meal based on the sequence $S$. The function $E$ can have very different forms depending on the individual and there are additional constraints depending on the situation.  Do you have to eat every items in $F$, or do you eat until you are full and put the leftovers in a doggie bag?  Is there a time constraint, in which case certain foods can be more easily digested and thus enjoyed more?  This problem can also be defined in more complicated ways by allowing you to eat part of one food item before eating part of another food item, etc.  $E(S)$ can simply be $E(S) = \sum_i E_i(s_i)$ where $E_i$ becomes smaller for larger $i$ as you are getting full, i.e. for all $x\in F$, $E_i(x)< E_j(x)$ if $i>j$. My argument at the beginning of this post was that $E_i$ decreases faster proportionally as $i$ increases for my favorite food than for foods I dislike.  To complicate things further, the ordering of the food items can influence your enjoyment not simply because your stomach is getting fuller, but could also depend on the flavor involved. Eating something sour might increase your appetite, whereas according to my wife eating something sweet after eating something salty might make you want to eat more salty foods. The use of sorbet to clean your palate between courses is another indication of this phenomenon. Similarly, eating protein or eating food that requires you to chew more will make you satisfied sooner. When presented with a plate of various fruits, she will eat the most sour fruits first, since she believes eating a sweet fruit first will make eating the sour fruit taste even more sour.
How different is the function $E$ different from person to person? Can we use our eating strategy $S$ over various meals to deduce the (presumably hidden) utility function $E$? (assuming that we are mentally optimizing $E$.)  Can optimizing $E$ be used to satisfy weight-loss goals?