Minority Game Applied on the Long Weekend

It was the longest long weekend in Australia last week – a total of 5 days holiday – it’s a combination of Good Friday, the weekend, Anzac Day and Easter Monday. There won’t be another 5 day long weekend in Australia until 2038. I had spent the long weekend in the Central Coast of New South Wales with my partner and made some very interesting observations.

We had left the house at about 8 a.m on Friday, to try to beat the traffic jam to the Central Coast. Feeling a little peckish, we decided to have some yum cha for brekkie. Alas, there were no shops within a 7 km radius that was open for yum cha. I live in a suburb with a fairly high concentration of Chinese food places, and none of the shops were open. That perhaps, would have been indicator of what was to come next.

We arrived at our destination at about 11 a.m, and feeling extremely hungry, decided to look for brunch. The whole town had only one eating place open. Like the yum cha places, most shops had decided to close for the long weekend. As I munched on my \$40 lunch (yes, it was a case of supply and demand – but that’s not the point of this article), I began to ponder upon the shops being closed.

To Open or Close?

What would cause shop owners to keep their shops open, or to close up for the weekend? As my friend Dylan would like to constantly remind me, it’s all about incentives, and expected utility. What incentives are there to keep the shop open? What incentives are there to take off for a holiday?

The answer to the former is simple – utility is gained from financial gain. The answer to the latter is also fairly simple – non-financial utility (i.e. taking a break). So, to keep the shop open or close is a simple matter of deciding which action provides more utility [1]. If it is more rewarding to keep the shop open than it is to take a holiday, then the shop will remain open during the holiday; and vice-versa.

So far so good – basics of game theory covered (to recap: if it’s more rewarding to do A over B, do A).

The Utility Functions

As I sat there thinking more and more about what would cause a shop to remain open during the holidays, I realized I must dig deeper into the utility functions [2] of  the shop owners to figure it out.

Take Shopkeepers Sara [3] and Oblomov. For simplicity’s sake, let’s assume that the holiday is one period. Their option sets, A are to keep the shop open, or close the shop and take a break from serving food. They each choose an option a based on their personal utility.  We can represent that mathematically:

$A_i = {open, close}, i = {Sara, Oblomov}$

$a_i = begin {cases} open & text{if } u_i(open) > u_i(close), , \ close & text{if } u_i(close) > u_i(open).end {cases}$

Now, let’s assume both parties have utilities that can be mapped on to real values. Let’s also assume, for simplicity’s sake, that both Sara and Oblomov value their holidays with the exact same utility, and their utility is equivalent to \$1000 in profit. This makes it a little simpler, as now their decisions whether to remain open depend on how much money they expect to make over the long weekend.

How much money would they make over the weekend would depend on a few things, which I’ve classified as internal and external. Internally, how much money a shop could make would depend on cost of running the shop, ability to draw customers (good food, good marketing etc). It would also depend on external factors, like the amount of substitute and/or complementary goods out there in the local market – for example, a cafe will be competing directly with another cafe, but a dessert shop would work well alongside a takeaway place.

Shopkeeper i‘s utility is the profit from running the shop, and profit is simply Revenue – Cost. Assume that Revenue is a function of number of people who visit the shop; and the number of people who visit the shop is a function of how much competition there is out there. So, we can now say that profit of a shop is a function of the amount of competition there is.

This is a fair assumption too, if you think about it. A cafe owner will make insane amounts of profit if she opens when every other cafes are closed (i.e a monopoly, albeit a temporary one). Her profits will drop as soon as competition begins to crop up. As other cafes begin to open, her profits might drop to a point where she gains more utility by taking a holiday instead of opening shop.

Let’s say that in a situation like the Easter long weekend, the audience is captive – i.e. the customers are not going anywhere (this particular resort town in the Central Coast does very well to keep its tourists in), and there is a constant spending pool that these customers have – that is to say every customer is willing and able to spend X dollars and the spending pool is the sum of the customer’s spendings. The spending pool can then be thought of as a pool of money to be shared out amongst the shop owners.

So, to restate this (π is profits, R is revenue, C is cost),

$u_{i}(open) = pi_{i} = pR - C_{i}, \ text{ arbitrarily, let }) p = frac{1}{n} text {where n = number of competing shops}$

This has become a situation where being in the minority would yield the most utility. And why does that sound so familiar? Oh yes, the minority vote (cue music and big words that say 少数決 and shake the screen a few times[4]).

The Minority Game

In 2007, cfgt and I stumbled upon an excellent manga series called Liar Game. Later that year, we found out that the J-drama adaptation of Liar Game was being aired. A featured game in Liar Game was the Minority Vote. In this game, a group of n players will vote a binary vote on a topic (yes or no), and the losers are those on the majority. Intrigued by this game, we set out to discover if there were any equilibriums to this game. We rather quickly came to a conclusion that there was no equilibrium in a pure strategy mode.

In 1992, W. Brian Arthur of the SFI [5] posited the El-Farol Bar Problem, which is a similar thing.  The El-Farol Bar Problem goes something like this: A bar is only fun when its occupancy is up to 60%. That is to say, if 60% or less of the population goes to the bar on a given night, they’d be better off than staying at home. However, if more than 60% of the population goes to the bar, a player is better off staying at home.

Both these games and the Easter Long Weekend Open/Close Conundrum are a class of games increasingly known as the Minority Game. There are other minority games in other forms, but I specifically chose the Liar Game and the El Farol Bar Problem examples because I would want to refer to them later. In particular, I modeled the minority game in a computer simulation, and several interesting things have shown up. The results and discussion following it will be discussed in the sequel article.

1. [1]If you’re lost so far, utility is a measure of relative satisfaction/happiness/rewardedness
2. [2]That is to say, it is assumed that everyone has von-Neumann-Morgenstein properties of preferences (completeness, transitivity, independence and continuous), and that everyone has a mappable utility function that allows a mapping of preferences to utility
3. [3]That was indeed the name of the shop owner who opened her cafe during the long weekend
4. [4]If you didn’t get the reference, it’s from a Japanese TV series called Liar Game
5. [5]If you have been reading this blog since its previous incarnations, you’d know that I’d love to work at the Santa Fe Institute when I have the correct qualifications