jwilzewske: Chamber Music Economics: Divisibility of Time and the Monty Hall Problem

I don’t know that there’s clean evidence that merely being asked to choose between two objects will make you devalue what you didn’t choose [in Monty Hall ‘Let’s Make a Deal’ choice situations]. I wouldn’t be completely surprised if this effect exists, but I’ve never seen it measured correctly. The whole literature suffers from this basic problem of acting as if Monty’s choice means nothing.”
— M. Keith Chen, quoted in John Tierney, And Behind Door No. 1, a Fatal Flaw, NYT, 09-APR-2008.

Most of the assumptions of normal mathematical models of exchange economies are violated in the case of arts markets, including live chamber music. The goods (the performances) are indivisible, yes, but the consumers’ economic ‘utility functions’ are nonlinear as well. The money that is used to pay for attendance is perfectly divisible, yes, but the time needed for attendees to consume the goods is only coarsely divisible (the performances are scheduled few and far between, and are often 90 minutes to 120 minutes or more in length), this in a competitive landscape where other goods’ consumption times are short, interruptible, and selectable ad lib. Thank you, Apple and iPod.

Despite the violation of assumptions that simplify mathematical analysis, live chamber music does constitute an exchange economy and does generate what economists call non-negative, superadditive transferable-utility (TU) games. As such, economic TU games usually exhibit price equilibria, in which case the TU game is said (mathematically) to be ‘balanced’. Conversely, if a TU game satisfies the mathematical criteria of ‘balance’, then it generates an exchange economy in which a price equilibrium exists. A recent paper by Meertens at the University of Nijmegen shows how this is so.

In fact, the ‘superadditivity’ property is a natural correlate of the at-best coarse divisibility of each person’s time. The time interval occupied by a concert may be considered to be the sum of smaller chunks of time. And the implicit total price or ‘opportunity cost’ of attending the concert is not merely the sum of the prices of the smaller chunks. The implicit total price is not less and, in general, is greater than the sum of the prices of the smaller chunks with which the concert competes. The whole is greater than the sum of the parts. Never mind the cash price of the concert ticket. That does matter, as to whether a person will attend or not and as to whether a price equilibrium for ticket prices will exist. Of course it matters. But a far larger component—one that’s been undervalued as a driver of the classical music market—is the value of large chunks of consumers’ time. None of the market research on this—the RAND / Knight Foundation studies, and many others—none of it has considered time as an economic variable.

Consider an exchange economy e with a finite set I of agents (whose elements are denoted by i, j, ...), a finiteset R of indivisible objects (whose elements are denoted by α, β, γ, ...), and a perfectly divisible good called money. Agents’ preferences are quasi-linear: the utility that agent i ε I derives from consuming a set of objects A can be characterized by a reservation value V(i, A) which represents the quantity of money that agent i is ready to sacrifice in order to consume the objects in A. The utility of agent i holding e_i units of money and the set A of objects is thus For all i ε I, the reservation value function V(i, .), defined on the set P(Ω), is assumed to be weakly increasing [V(i,A) ≤ V(i, B)] and to satisfy V(i, 0) = 0. Agents’ endowments, (Ā_i, ē_i), with ē_i ≥ 0 are assumed to be such that ē_i V(i, Ω), for all i . This assumption implies that whenever the price of a set A of objects is less than the reservation value V(i, A), agent i can afford to buy the objects in A.

An ‘efficient’ allocation or assignment σ of the goods to the consumer-agents must satisfy this equation, according to Beviá and colleagues:

A price equilibrium exists if the agents have a non-zero utility for at most one good or if all of the choices are identical. But what happens if the goods are not equal and perfect substitutes for each other? Beviá and colleagues studied the situation where the marginal utility of an additional item in a “bundle” of items decreases when the bundle of goods to which it is added gets larger. This is called ‘submodularity’. They studied the following simple example, involving 3 consumer-agents and 3 goods.

	α	β	γ	αβ	αγ	βγ	αβγ
V(1,A)	10	8	2	13	11	9	14
V(2,A)	8	5	10	13	14	13	15
V(3,A)	1	1	8	2	9	9	10

The only efficient assignment σ of goods in this economy is such that σ(1) = β, σ(2) = α , σ( 3 ) = y. Suppose that p supports this assignment. In order that buying { α β } is not better for agent 2 than buying only α, p must be such that p(β) ≥ 5. In order that saving the money and buying nothing at all is not better for agent 3 than buying γ, p must be such that p(γ) ≤ 8. In order that buying α is not better for agent 1 than buying β, p must be such that p(α) ≥ p(β) + 2. In order that buying γ is not better for agent 2 than buying α, p must be such that p(γ) ≥ p(α) + 2. Combining these inequalities gives 7 ≤ 2 + p(γ) ≤ p (α) ≤ p(γ) - 2 ≤ 6, which is mathematically impossible. So this 3 agent - 3 object example of an exchange economy is one where the optimal assignment of goods is not supported by a a price vector. The TU game exists and the exchange economy e exists, but a price equilibrium does not exist (with the utilities and reservation values / preferences, for these particular agents).

If agent 2 has objects α and β, then the marginal contribution of α is equal to its value V(2, α) since V(2, αβ) - V(2, β) = 13 - 5 = 8 = V(2, α), while if objects α and γ are combined the marginal contribution of α is much lower: V(2, αγ) - V(2, γ) = 14 - 10 < V(2, α). For agent 2, having object β at the same time doesn’t subtract any of the value of α while having γ lowers the desirability of α. So the exchange economy in these objects exhibits this funny type of inter-dependence. What you have chosen matters. (You are acting, in your chamber music consumption patterns, as if what your significant other, Monty, has chosen does mean something?)

Well, not exactly. This type of inter-dependence is unrelated to the Monty Hall Problem and its different type of interdependence. In Beviá’s example, money is infinitely, perfectly divisible. And the example implies that there is no good that was demanded by agent i by reason of its being particularly well-suited to accompany a good that agent i was already committed to owning and consuming, but instead implies that there are goods that agent i does not demand because the good that agent i already is committed to has become more expensive. Agent i declines to attend a chamber music concert, not because the ticket price and parking price are too high, but rather because of the time-cost in the context of existing commitments/choices.

In this regard, chamber music presenters should consider that ‘time is money’, as they say. Each potential concert-goer’s time is infinitely divisible, just as money is infinitely divisible. But a concert experience represents a considerable implicit time-cost, on account of the event’s duration, plus the durations of all of the logistical actions that are collateral to attending (driving, parking, etc.). As such, attendance at a concert—even a ‘free’ concert—carries a high and, over the past 10 years, increasing time-cost, relative to the other goods in the entertainment and cultural economy with which concert attendance competes.

There have been experiments—many of them over the past 10 years or so—in changing the length and format of classical music concerts. None so far has been particularly successful, and some have failed miserably, enraging long-time patrons who booed or revolted when their treasured, time-honored ‘good’ had, in their view, been tampered with beyond all recognition. So the aim of the game-theoretic ideas above is not to suggest that concerts need to be shorter or more numerous or scheduled on different days-of-week or times than is presently being done.

No, the aim of acknowledging recent developments in game-theory and mathematical economics modeling in this post is not to recommend game-theoretic analysis as a source of new, quantitative ways to address the situation that chamber music presenters (and classical music performers and agents, and the arts in general) find themselves in today. The aim instead is merely to say that game-theory can explain many aspects of why things are the way they now are.

There is little solace in this, I admit. Except, perhaps, the realization that the same long-duration time-indivisibility and tee-time quantization and ‘money-value-of-time’ factors have also led to progressively declining annual golf ‘rounds-played’ stats, as documented by the National Golf Foundation, over approximately the same post-internet time interval that has affected classical music events. The same factors are affecting the economic regimes of other long-duration event markets as well. Classical music is not alone in this.

Solace or not, the analyses by Beviá and Bikhchandani and Meertens and others do provide a rational basis for chamber music presenters’ not pursuing strategies that have little hope of succeeding in building audience and cashflow, and do provide a methodology for properly conducting analyses of new proposed strategies that come from brainstorming and other sources, before investing in and committing to those strategies.

jwilzewske

Thursday, April 10, 2008

Chamber Music Economics: Divisibility of Time and the Monty Hall Problem

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