Game Theory and essay choosing and writing *

2009-06-26 9:58 p.m.

I’m putting this entry up again—with little editing. I’ll edit it later….

Let us assume there are two students, A and B (I could give names but let me be conservative and simple). They are given a choice of two essay questions Q1 and Q2 these are the only questions available. In the library, there are a number of books, half of which (detailed by the letter x) are related only to Q1 and another half (detailed by the letter y) related only Q2. Both questions are of equal difficulty and both students are given 10 weeks to complete one essay of either question.

The simple two player game arrives by the choice—choosing which question. (At this point let me say I’m not drawing the boxes because I just can’t figure out how to post them). This game is the simple two player two choices model. So to cut the numbers and permutations short, we know the best choice is for A to choose Q1 and B to choose Q2 (or vice versa) so they can get the same number of books from the library. Is it possible to achieve a least utility payoff—ie the utility (score) of both players is low or close to zero? If both suddenly go for the same question and they have a hard time going exchange books, their essays may turn out horrible.

Well given the possible choices, naturally there is the choice of both going for the same question. The definition of the Nash Equilibrium—where each player’s strategy is a best reply to the other—comes into action. If you imagine the boxes (or just drawn them out with numbers) there would be at least a Nash Equilibrium choice—although one student may have a higher utility there—ie. One reads the book(s) faster, comprehends it easily and can get down writing than his or her opponent. So there are Pareto optimal points for both players. Which of course leads us to the next famous topic in game theory—The Prisoner’s Dilemma.

Students A and B, as you might have guessed, have so far been playing simultaneous games so far. There may be a really perfect combination such that both can produce great essays and gain much satisfaction out of it. However, support A chooses Q2 but B’s best choice is Q1 (maybe his ability to gather the x number of books is easier). If A chooses Q2 but B’s best reply is also Q1 (for such and such a reason), there is a dominant strategy. If is works out similarly for A and to cut the whole story short, there is best possible choice for both players, but it may not be the most Pareto optimal.

What if one student moves first? Suppose both want Q1 but B is busy. A goes for x number of books but B makes a deal so that he/she gets those books exactly after five weeks (the halfway mark). In that case, with all other factors being equal, both have access to the same number of books, although their essay speed and quality may be different. Or A plays first and B plays second but with both wanting Pareto optimal outcomes. Here’s it is that of mixed strategies. In fact, the game is more complex now with mixed strategies as a probability set can be formed—the probably that A chooses Q1 and Q2 and the probability that B chooses Q1 and Q2. With some mathematics, you can get each student’s Nash Equilibrium payoff.
Let us bring this closer to reality. Suppose midway through the A-B deal mentioned above, A breaks the deal (just like the breaking of a collusion) and hogs all the x number of books, preventing B from attempting Q1. With five weeks or so to the assignment deadline B loses out. Or suppose A returns a portion of the x books but the library suffers a disaster. A can only give B a certain number of books then. Or cooperation could take place where A and B go for the same question but share books together. Even so the “breaking of the pact” can still occur. Should a collusion be planned, both A and B need to plan collusion strategies.

Ok, let’s throw the case open to the real world. Firstly, library books are not the only source of information for essays. So if your fellow student cheats on your by hogging books, you at least have several other alternatives. Secondly, assignment questions are never of the same difficulty. However, assuming students are rational decision makers, students would go fro the easiest question. Game theory however, would show that your utility may not be maximised by going for the easiest. The other most evident point is that there will be more that two questions to choose from and more than two students. Even so, Game theory still applies—which set of combinations is the most favourable? Will students collude when sourcing for their information? Where are the Nash Equilibriums? Another lot of factors are that of student’s time that can be spent on the essay, the ability to gather the books or other materials.

So, is essay writing that easy? Think again.

N.B. I hope I got most of the game theory stuff right. What has been shown is of course simply Prisoner’s Dilemma and Nash Equilibrium—the basics which you can pull out from a Microeconomics textbook. There are other related games which still apply to essay question choosing. Credit of my learning of game theory goes to Dr. Vasilakos, whom I felt I let down but who made the learning of this topic a very enjoyable one—especially when he used the example for “A Beautiful Mind”.

Please DO NOT reproduce this entry elsewhere without my permission. In fact, I may take it down after a while. For more details, contact me.

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