Rationalisation

Activity (20 minutes)

Goal. Let students feel the best response logic before we formalise it.

The deck. There is a write-on deck at /decks/rationalisation/main.pdf. It carries the game, a table for the six rounds played in pairs, and the three-mix table that used to be a separate sheet, so nothing else needs printing.

First, in pairs. Pair the students up and have them decide who is the row player. They play two games, six rounds each, choosing at the same time, and fill in every column: both actions, both payoffs and both running totals.

The first game is matching pennies, so the payoffs are mirror images and the table is easy to fill. The second is the asymmetric game we actually want, where the cells no longer mirror and they have to read each one. Doing the easy one first means the second table tests their reading rather than their nerve, and the six rounds are the same six they then play against the dice.

Then against the dice. The row player is now a die with a known mix. Before each roll a student commits to \(c_1\) or \(c_2\); then roll, read off \(r_1\) or \(r_2\), and score. Six rounds for each of three mixes:

One D12 per pair covers all three, which is why the mixes are quarters and thirds rather than the tenths we used to use. Twelfths divide exactly by both, so each probability is exact rather than approximated. The three are chosen so that one sits below the indifference point, one above it and one exactly on it, which is the whole point of the exercise. Note that no single standard die can produce \(0.2\) and \(0.9\) alongside \(1/3\): that would need a multiple of 30 faces.

Because the students roll their own dice, the mixes are printed on the deck rather than hidden. The commitment still matters: they must write down their \(c_1\) or \(c_2\) before each roll, so the running total is a fair test of the reply they chose rather than of hindsight.

Discussion (20 minutes)

Discuss the Rationalisation chapter.

Discussion Point: After the definition of dominance, ask why dominance is not relevant to the game we played.

Discussion Point: How does the best response condition apply to the matching pennies game we played in class?

Answers for the deck

Headings match the deck's page numbers, so this can sit open on a second screen while the deck is projected.

Page 3, matching pennies. Zero sum: every cell adds to \(0\). Neither action is dominated, and there is no equilibrium in pure actions, so anyone who always shows the same thing loses to an opponent who notices. The equilibrium is both players mixing \((1/2,\ 1/2)\), with value \(0\).

Page 4, the main game. Also zero sum, every cell adding to \(0\). If the row player picks \(r_1\) and the column player \(c_2\), the column player scores \(+2\) and the row player \(-2\).

Page 5, the dice. One D12 per pair.

Page 6, working it out. Against \(\sigma_1 = (p,\ 1 - p)\),

So \(c_1\) is the better reply when \(p < 1/3\), \(c_2\) when \(p > 1/3\), and the two are equal at \(p = 1/3\).

Page 7, dominance. Neither action is dominated, because \(c_1\) is better for \(p < 1/3\) and \(c_2\) for \(p > 1/3\). Dominance only rules out an action that is worse against every strategy of the opponent, and both of these are a best response to something, so the best response condition is what we need.

Page 8, indifference. The third die pays \(0\) whatever the column player does, so their six-round total should hover around \(0\). Their own mix that does the same to the row player is \((1/2,\ 1/2)\), which gives the row player \(0\) from either row. Together with the row player's \((1/3,\ 2/3)\) that is the Nash equilibrium, and the value of the game is \(0\).

What to watch for in the pairs rounds. Nothing to compute here, so watch instead for pairs who settle into one action and get punished for it, and for pairs who start alternating. Name both in the debrief: the first shows why a pure strategy is exploitable, the second is the room inventing mixed strategies before we have defined them.

From the activity to the exam answer

This activity feeds into the Nash equilibrium material. Use Question 1 (the in-class activity) on the Nash Equilibrium page to show students how the ideas here become a full-mark exam answer.

Communications

General email templates to send before and after this class. Fill in the bracketed placeholders before sending.

Before class

Hi all,

A reminder that our next Game Theory class covers Rationalisation.

All of the course materials, including the relevant chapter, are available
at https://vknight.org/gt/. It is worth skimming the chapter beforehand.

See you in class,
Vince

After class

Dear all,

Thanks for your work in today's class on Rationalisation.

A recording is available here [RECORDING LINK] and on Learning Central.

All class resources are available at https://vknight.org/gt/.

Thanks,
Vince