Chapter 1: Games and Decisions

7 Oct

This is my first real post. I tried to express in my introductory post that my intention in this blog was to challenge myself to look Institute and Faculty of Actuaries’ Core Technical material in a way not necessarily suggested by the material itself.

The first chapter of CT6 Statistical Methods is a brief look at Game Theory and Decision Theory.

In this chapter some of the essential terminology of these two related topics is introduced.To wit:

  1. Dominated strategies
  2. Maximin criterion
  3. Saddle point strategy
  4. The Bayes criterion

I think the authors’ of this part of the notes which is that the opportunity to use whimsy in the examples – one of the best opportunities throughout the Core Technical series, given it is hard to find spice in interest theory calculations. In this regard, I recommend Luce and Raiffa’s Games and Decisions (1957) available from Dover for no more than semi-drinkable bottle of wine (in Australian bottle shops, anyway. Depending on the country you may get anything from a completely undrinkable to really quite decent for that money). Note though that out of fourteen chapters, only two (chapter 4 and chapter 13) really correspond to the material in CT6, though.

While the above objection is only half serious, another way in which the Luce and Raiffa treatment is more interesting, is that for each classification of game it discusses, it gives an example game either of research interest (whether for theoretical or practical reasons). In particular the egg craking story ( borrowed from Savage) used to motivate statistical decision theory is a far better illustration that a statistician tossing a coin used in CT6. Although, as a married man, I am a little distracted by the questions thrown up by the opening sentence of this example – ‘Your wife has broken five eggs into a bowl when you…volunteer to finish making the omelet’. The problem is to decide whether or not to check if an egg is rotten before craking it into the bowl and either making a large omelet or ruining five eggs.  If it was not your wife, but you girlfriend, daughter, kitchen hand (to your chef) or housemate, how would your decision process change?

In the end there is a purpose to fitting game and decision theory to real life situations, so to redress the lack in the CT6 notes I offer an inversion of the problems suggested there based on TV’s The West Wing.

In the final series, a Republican and Democratic presidential candidate are in the race to become Bartlet’s successor. In the episode Duck and Cover a nuclear power plant whose approval was made possible by the lobbying of the Republican candidate goes into meltdown, and at least one of the repair crew dies after trying to fix it. The respective campaign managers must decide whether to put out a statement telling the world about the Republican candidate’s role. For the Republican candidate, putting out a statement first could minimise the political damage because their side of the story will get told first, but if there is no statement from the Democrats, it will reveal the existence of the connection. On the other hand, if the Democratic candidate points out the connection to the press, it could either be helpful to their campaign if t harmful depending on whether there is a Republican press statement available to neutralise it. Propose a pay-off matrix which describes this situations. Is there a spy proof strategy?

The example above operates under the rules of decision making under certainty. In the show, there is an element of uncertainty in the form of whether the press discovers the connection and when. Adding this element to the problem above makes for a more complicated example.

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