The toilet seat game

When I blogged my review of William Nicolson's "The Romantic Economist - A Story of Love and Market Forces" last year, I mentioned that I would definitely use the toilet seat game as an example in ECON100 this year. However, the game theory topic is already pretty full, so I didn't manage to squeeze it in. So instead, I thought I would talk about it here.

First, a little bit of background. William tells us about one of his relationships, with Sarah, which hit a bit of a rocky patch with respect to toilet seats, and whether they should be left up or down. You may laugh, but there's been several papers that have explored the game theoretical implications of the toilet seat game (see here and here for two examples).

In the case of Will and Sarah, the decision-making can be thought of as a sequential game, with three decision-making nodes. In the first node of the game, Will decides whether to leave the seat up or down. If he leaves the seat down, the game ends (because Sarah is happy). But if he leaves the seat up, Sarah gets angry. She then has the choice of whether to forgive Will, or punish him by nagging or throwing a minor tantrum. If Sarah forgives Will, then the game ends (and Will is pretty happy, because he got to leave the seat up and avoid the nagging). If instead Sarah punishes Will, then Will has the choice of agreeing to change his behaviour for the future, or ending the relationship. The game as a decision tree (extensive form) is presented below. The payoffs to Will and Sarah are simply ranked from 1 (best) to 4 (worst). Note that in this version of the game, ending the relationship is a worse outcome for Sarah than for Will. [*]

We can solve for the (subgame perfect) Nash equilibrium by using backward induction - essentially we start at the end of the game and work our way backwards, eliminating strategy choices that would not be chosen by each player. In this case, the last play is by Will. He can choose to end the relationship (and receive his 3rd best payoff), or change his behaviour (and receive his worst payoff). So, clearly he would choose to end the relationship. Now, working our way back to Sarah's choice, if she punishes Will then we know that Will is going to end the relationship. So Sarah can choose to forgive Will (and receive her 3rd best payoff), or punish him (which results in Will ending the relationship and Sarah receiving her worst payoff). Given that choice, Sarah will choose to forgive Will. Working our way back to the first node then, Will can choose to leave the seat down (and receive his 2nd best payoff) or leave the seat up (knowing that Sarah will forgive him, and he will receive his best payoff). Given that choice, Will is going to leave the seat up. So, the subgame perfect Nash equilibrium in this case is that Will leaves the seat up, and Sarah forgives him.

Will then goes on to point out that the outcome of this game crucially depends on how Will feels about Sarah. If he really wants to be with Sarah, then the game changes. Now say that ending the relationship is the worst outcome for both Sarah and Will, as showing in the decision tree below.

Solving for equilibrium in this case (again using backward induction), we find that Will would agree to change his behaviour (because that provides him with his 3rd best payoff, compared with the worst payoff if he instead chose to end the relationship). Now, knowing that Will is going to change his behaviour, Sarah would choose to punish him if he leaves the seat up (because she would receive her second best payoff when he agrees to change, which is better than her third best payoff, which she would receive by forgiving Will). Now, knowing that Sarah will punish him, Will decides to leave the seat down (and receive his second best payoff, which is better than the third best payoff that he would receive if he left the seat up, because Sarah would then punish him and he would then change his behaviour).

All in all, this is a great example of a sequential game in action. Of course, the game as presented above is necessarily simplified - in fact this game is a repeated game, so the Nash equilibrium is more complex (as shown in the two papers linked at the beginning). But a great example nonetheless.

*****

[*] Careful readers of Nicolson's book will notice that I have altered the payoffs in these games from those that appear in the book. Specifically, the payoff for Sarah in the case of the Up/Punish/Change outcome is that this is Sarah's second best payoff (whereas in the book it is her third, which is equal to the Up/Forgive outcome). I expect this was a slight error in the book.

Uber says they're not price discriminating - should we believe them?

James from my ECON100 class pointed me to this article about Uber, surge pricing and price discrimination:

The car-hailing service Uber can detect when a user’s smartphone is low on battery, and therefore willing to pay more to book a ride.  

Uber, which has faced the ire of London’s tax drivers since launching in the capital in 2012, can tell when its app is preparing to go into power-saving mode, although the firm says it does not use this information to pump up the price.

Keith Chen, head of economic research at Uber, told NPR that users are willing to accept a “surge price” up to 9.9 times the normal rate, particularly if their phone is about to die.

“One of the strongest predictors of whether or not you’re going to be sensitive to surge… is how much battery you have left on your cellphone,” he said.

“We absolutely don’t use that to push you a higher surge price, but it’s an interesting psychological fact of human behaviour.”

So, Uber has discovered that people are more willing to pay a higher surge price when their phone battery is about to die (see my previous post on surge pricing). This suggests that Uber passengers have less elastic demand when their phone is running low on charge. This makes sense for a couple of reasons, both relating to time horizons. First, if you need to get home (or to work, or to a meeting, etc.) and your phone is running low on battery, then you need to get a ride soon. If you wait too long, your phone might run flat and then you might find it more difficult to find a ride home (you'd need to resort to hailing a cab on the street, using public transport, or walking). Second, we don't want to be without our phones for too long, lest we miss Kim Kardashian's latest tweet. So, we want to get somewhere where we can recharge our phones, and fast. We know that when customers have short time horizons, their demand is relatively less elastic. And when demand is relatively less elastic, firms can mark up the price higher and the customer will still be willing to pay.

So, should we believe that Uber is not price discriminating? Price discrimination increases profits when firms can do it effectively. This only requires three conditions to be met:

1. Different groups of customers (a group could be made up of one individual) who have different price elasticities of demand (different sensitivity to price changes);
2. You need to be able to deduce which customers belong to which groups (so that they get charged the correct price); and
3. No transfers between the groups (since you don't want the low-price group re-selling to the high-price group).

The first condition is clearly met, and presumably Uber's app knows when the phone is low battery (it's probably buried in the terms and conditions for the app, which almost no one reads). Since customers don't know the battery status of other Uber customers, then the third condition is likely to be met too. So, if Uber isn't price discriminating on the basis of battery level, they are leaving potential profits on the table. Uber shareholders probably wouldn't be too happy to learn this. So, I think it's hard to believe that Uber don't take a lot of information about their passengers (including potentially the remaining battery life of their phone) into account at least at some level - perhaps they are not price discriminating via the surge price (i.e. the multiple by which they increase prices), but via the underlying base price?

As an interesting side-note, later in the article we find out a bit more about the price elasticity of demand of Uber customers when surge pricing is new, and then later:

When Uber first introduces surge pricing to a city, even a small jump in price to 1.2 times the normal rate is enough to discourage 27pc of potential customers from booking.

However, cities with more established Uber services see a smaller 7pc reduction when this price rise kicks in, showing customers get used to the idea, Chen said. 

This suggests there is relatively elastic demand when Uber first introduces surge pricing to a city (since the percentage change in quantity (27%) is greater than the percentage change in price (20%)). But there is relatively inelastic demand once customers are more used to it (since the percentage change in quantity (7%0 is less than the percentage change in price (still 20%)). This suggests that people are initially resistant to temporary increases in price (through surge pricing), but once they realise the benefits of it, that initial resistance fades.

I also found this bit interesting:

Mr Chen also noted the quirks of psychology that make passengers more likely to ride when the price is 2.1 times normal, rather than 2.0 times, which chimes in their consciousness as being twice as expensive.

“Whereas if you say your trip is going to be 2.1 times more than it normally was, wow, there must be some smart algorithm in the background here at work, it doesn’t seem as unfair,” he said.

I've seen other research that supports this (but off the top of my head I can't recall where) - it showed that people are more likely to be confident in a forecast share price of, say, $2.02 per share, than a forecast of $2.00. People believe there must be some sophisticated underlying model for a forecast of $2.02, whereas $2.00 seems more like a guess. Aren't cognitive biases wonderful?

Why study economics? Law and economics edition...

A recent paper (ungated here) in the Journal of Economic Education by John Winters (Oklahoma State University) looks at the earnings of lawyers in the U.S., by their undergraduate major (note for New Zealand readers: in the U.S., law requires graduate-level study, so all law students would have previously studied an undergraduate degree first). John explains:

I build on previous literature by presenting new data on lawyer earnings by undergraduate college major. Specifically, the data are from the pooled 2009–13 American Community Survey (ACS), which provides a 5-percent sample of the U.S. population. I report both mean and median earnings, and find that economics majors have especially high earnings among practicing lawyers.

Interestingly, economics is the fourth most common undergraduate major (studied by 6.5% of lawyers in the sample), behind political science and government (21.6%), history (9.9%), and English language and literature (8.1%). However, economics outperforms all of those more common majors in terms of earnings:

Lawyers with undergraduate majors in electrical engineering have the highest earnings according to both medians ($179,744) and means ($219,383)...

Accounting majors have the second highest median ($135,044) and third highest mean ($180,507) earnings. Economics majors have the third highest median ($130,723) and the second highest mean ($182,359) earnings.

Political science and government ranked ninth in median earnings, history ranked eighth, and English language and literature ranked 15th. So of the four most common undergraduate majors among lawyers, economics came out on top.

Law and economics are very complementary - for instance, Wikipedia has a page devoted to law and economics, and the late Nobel prize-winner Ronald Coase was famously Professor of Economics at the University of Chicago Law School. Top economists Gary Becker (the 1992 Nobel prize winner) and Andrei Shleifer have made seminal contributions to research in law and economics.

At Waikato we have taught undergraduate and graduate papers in law and economics for many years, and some of our top graduates have completed conjoint degrees that combined a law degree with either a management or social science degree majoring in economics. Knowing the great jobs that those graduates have gone onto, I wouldn't be surprised if the earnings for law and economics graduates was higher than many other combinations for our graduates as well.

Add this to the list of reasons to consider including economics in your degree programme (if you're a current or future law student).

Read more:

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• Why study economics? Tech firm jobs edition...
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• Why study economics? Even if you won't be an economist edition...
• Why study economics? PhD edition...
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• Why study economics? Sheepskin effects edition
• Why study economics? NZ graduate earnings edition
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Why block pricing doesn't work for heterogeneous demand

On Sunday I covered two-part pricing. Today it's the turn of block pricing. A firm uses block pricing when it charges a relatively high price until the consumer reaches some threshold, then a lower price for units after the threshold. In reality, there could be multiple thresholds. Buy-one-get-one-half-price is an example of block pricing. As with two-part pricing, we can think about block pricing first by contrasting it with a monopoly firm pricing at a single price-per-unit, as in the diagram below (for simplicity, I'll again use a constant-cost firm).

The monopoly firm using a single price-per-unit selects the price that maximises profits. This occurs where marginal revenue is equal to marginal cost, i.e. at the quantity Q1 with price P1. The producer surplus (profit) the firm earns is the rectangular area CBDF.

However, if the firm switches to block pricing, then the firm can choose an initially higher price (P0), at which point the consumers purchase Q0 of the good. If the firm lowers the price to P1 for every unit the consumer buys after Q0, then the consumer will buy Q1 of the good (and pay P0 for the first Q0 units, then P1 for the rest). The firm's profits would increase by the area HGJC.

The firm could block again, offering a lower price than P1 for units purchased beyond Q1, and capture more profits (I haven't shown this on the diagram though). Again, this demonstrates that firm profitability is all about creating and capturing value.

We can also show the effect of block pricing using the consumer choice model. This is illustrated in the diagram below. The black budget constraint represents the most the consumer can afford to buy with their income, when there is a single price-per-unit for Good X (with 'All Other Goods' [AOG] on the y-axis). The consumer purchases the bundle of goods E, which includes X0 of Good X, and A0 of All Other Goods.

With block pricing, the firm charges the standard price Px up to the quantity X0, then pays a lower price beyond that quantity. This causes the budget constraint to pivot outwards (and become flatter) from the point E. So this consumer can now reach a higher indifference curve, by buying the bundle of goods D (their new best affordable choice). This bundle includes more of Good X (X1), and less of All Other Goods (A1). Because they are buying less of All Other Goods, they must be spending more on Good X.

As with two-part pricing, block pricing also works well when the firm faces homogeneous demand for its product (i.e. when all consumers have similar demand for the product). We can also use the consumer choice model to demonstrate why block pricing doesn't work so well when there is heterogeneous demand.

Consider two consumers - one with low demand (shown on the diagram below with the blue indifference curve), and one with high demand (red indifference curves). With a single price-per-unit, the low demand consumer buys Bundle G, which includes X1 of Good X, and A1 of All Other Goods, and the high demand consumer buys Bundle J, which includes X3 of Good X, and A3 of All Other Goods.

When the firm moves to block pricing instead, the low demand consumer is not affected. The highest indifference curve they can get to is still I0, so they continue to buy Bundle G. 

The high demand consumer would be better off moving to buying Bundle K, which is on the highest indifference curve they can now reach. Bundle K contains more of Good X (X4), so block pricing does induce these consumer to buy more. However, Bundle K includes more of Good A (A4), which means that even though these high demand consumers are buying more of Good X with block pricing, they are actually spending less on Good X.

So, block pricing doesn't work so well for heterogeneous demand, because the lowest demand consumers will not be affected, while the highest demand consumers will buy more of the good, but spend less on it. The combination of these two effects is likely to reduce the firm's profit, but notice that it isn't as bad as for two-part pricing that I described on Sunday.

Again, this is why you don't often see block pricing (or two-part pricing) alone 'in the wild'. Firms must first ensure they have relatively homogeneous demand, and they achieve this through price discrimination (e.g. through menu pricing - offering different options to different consumers, knowing that each option will appeal to a different 'type' of consumer). Then within each homogeneous group they can use block pricing or two-part pricing.

Read more:

• Why two-part pricing doesn't work for heterogeneous demand