jwilzewske: game theory

Showing posts with label game theory. Show all posts

Thursday, October 16, 2008

Merge Two Competing Chamber Music Presenter Organizations?

T he effect of the economic downturn is already pronounced—in terms of our subscription sales and attendance so far this season, and in terms of corporate and foundation money. I wonder whether, if we merged with the other main chamber music organization in our city, we might do better overall. The reason I think the answer may be ‘yes’ is that our respective programs tend time and again to collide with and compete against each other for the same audience. For example, two pianists in one week--the other org’s program on Friday night, ours on Saturday night. Or two early music programs within a fortnight. The ‘supply’ [of chamber music programs] exceeds the ‘demand’ in our market area, or at least exceeds our audience members’ monthly budgets of time and money. CMT sometimes has spreadsheets and math [to illustrate how some process works or to provide a tool to help CMT readers’ decision-making]. Could you do something in Excel to show whether there would be financial advantages or disadvantages if we combined with our competitor? The assumptions would be that we would have the same number of events each season [Presenter P’s events + Rival R’s events]; the prices and expenses would be the same [P + R roll-up]; the staff would be the same [P + R, with executive co-directors and artistic co-directors]; and the corporate and foundation funding would be the same [P + R]. If we merged, we would just coordinate our programs to not compete—to more efficiently and effectively serve the demand in our community. Possible?”
— Anonymous.

The performing arts market is tremendously fragmented. That fragmentation inevitably leads to inefficiencies. There are more than 520 presenter entities who are organizational members of Chamber Music America. And one thing that’s clear from examining CMA’s directory of chamber music presenters is that communities in the U.S. that have performing arts markets that are robust enough to have one presenter tend in fact to have two or more chamber music presenters. In many cases, that means that there is relatively intense competition for what is almost certainly a finite market—a finite monthly or quarterly consumer spend per household. Probably the same is also true in cities in Europe and the U.K.

For simplicity and to directly respond to the anonymous emailer’s question, I’ve put together a mathematical model that is for two competitors in a market—a duopoly. It would be far more complex to create an accurate, actionable financial merger model for three or more competitors. Actually, if the proposition were to simultaneously consolidate three or more competitors into one unified presenter organization, then you could still use this Cournot-Nash game-theory model as-is. You would simply put your own figures in as Presenter P, and then sum the figures for all of your competitors and put those sums in the Rival R column.

Basically, you need the ticket sales (earned income) figures for you and your competitor for last season's events—not the ticket drop numbers (with comp tickets and other imponderables) but the cash money taken in. You can exclude the events that each of your orgs produced that did not compete with each other at all—because they were far enough apart (say, more than 4 weeks) so that it’s implausible that a potential audience member would’ve decided to decline to attend your event because they were already attending your competitor’s event, or vice versa. Then you adjust the up-down arrows so that the accomodation figures match your last-season historical values, and so that the Cournot-Nash duopoly figures on the left more or less match the last-season actual average per event period figures on the lower right. (Just click on either of the screen-shot images below to Open or Save_as the Excel spreadsheet.)

Program ‘event periods’ means any interval of time during which the competitors’ programs compete against each other for consumers’ dollars. It doesn’t have to mean conflicting events on the exact same dates. It may be events on adjacent dates, such that attendees who otherwise would like to attend both programs probably will not buy tickets and attend chamber music concerts on two consecutive days. It may be events during the same week or fortnight, with the same criterion that most members of the target market may not attend two or more chamber music events within, say, 10 days of each other.

There are a number of assumptions and limitations of this simple Cournot-Nash model of financial competition:

It doesn’t take into account the possibility of ‘curvature’ of the elasticity of demand;
It ‘linearizes’ the [possibly non-linear] competitive interaction;
It uses the statistical covariance cov(P,R) between the competitors as the measure of the ‘accomodation’ effect of the sales of one presenter on the competitor’s sales, which, while simple, may be a far-from-ideal metric of the competitive economic interaction between the two;
It doesn't account for potential greater-than-additive ‘synergies’ in terms of induced greater demand or brand-recognition or marketing effectiveness that a merged entity might achieve;

and so on. But for a basic, first-order model it does pretty well. It can give you realistic estimates of how strong or weak the competitive process is in your market. And it can give you reasonably accurate, quantitative guidance regarding how large or small the impact of merging/consolidating would be.

With the insights you glean from playing around with this simple model, perhaps you will try to arrange your programming timing and content so as to minimize the numeric value (covariance) of your own ‘accomodation’ to your competitors—i.e., select your artists and programs so as to make your own ticket sales very insensitive to the programming that your competitors present, while simultaneously maintaining your ‘brand’ and maximizing the demand for the programs you select and book.

This simple model can be used to devise other strategies: to make your organization attractive for a merger or, conversely, to make your organization an unattractive target (by removing any appearance of financial advantage associated with combining and coordinating programming so as not to compete). ‘Accomodation’ values that are large (> 40% for one or both competitors) tend to predict financial gains for a merged entity that are upwards of 30% compared to the total annual sales with each competitor separate. Conversely, ‘accomodation’ values that are low (< 10% for one or both competitors) tend to predict that merging the competitors would not net much income growth for the merged entity—growth of 15% or less.

So please have a look at the model. Send me email or comment on it if you wish. And give us your thoughts in the poll that’s embedded in this post. (Note: Your participation in the online poll does not disclose your own identity or your organization’s identity, nor does it reveal anything about your community. It does not collect information other than which selection you click on.) Thank you!

Online Surveys & Market Research

Thursday, April 10, 2008

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.