Complex Adaptive Systems and the Threshold Effect: Views from the Natural and Social Sciences: Papers from the AAAI Fall Symposium (FS-09-03) Dynamics of Price Sensitivity and Market Structure in an Evolutionary Economic Matching Model Griffin Vernor Drutchas and Péter Érdi Center for Complex Systems Studies (CCSS) Kalamazoo College 1200 Academy Street Kalamazoo, MI 49006 [email protected] [email protected] Abstract ward these related public and private ends, creating a simu- The relationship between equilibrium convergence to a uni- lated laboratory allows safe exploration of interdependency form quality distribution and price is investigated in the Q- and uncertainty, system dynamics, and potential policies model, a self-organizing, evolutionary computational for improving market outcomes. matching model of a fixed-price post-secondary higher edu- cation system created by Ortmann and Slobodyan (2006). Building on the Q-Model The Q-model is replicated with price equaling its Ortmann and Slobodyan (2006) value P*. Varying the fixed price be- The following model and simulation experiments build tween 0 and 2P* reveals thresholds at which the Q-model upon the self-organizing economic models of Nicholas reaches different market clustering configurations. Results Vriend (1995) and Andreas Ortmann and Sergey indicate structural market robustness to prices less than P* Slobodyan (2006) in creating a decentralized market “cul- and high sensitivity to prices greater than P*. ture-dish that allows [one] to explore how macro regulari- ties emerge through the repeated local interactions of boundedly rational, heterogenous agents” (Ortmann and Introduction (1) Slobodyan 2006). Vriend (1995) builds and explores the Over the past two decades, the American post-secondary baseline dynamics of an evolutionary economy where ho- education system—the collection of colleges, universities, mogenous firms and consumers seek transactions while and institutes that offer education beyond high school—has behaving based on the success of past choices. Ortmann expanded to accommodate growing student enrollment. and Slobodyan (2006) generalize Vriend (1995) into the Q- From 1985 to 2005, full-time attendance at post-secondary model: a firm-consumer matching model that includes het- institutions rose 33 percent (NCES 2007). Attendance erogenous agents stratified along a quality spectrum and is rates were higher still for historically underprivileged applied to the study of the post-secondary education sys- groups. Nonetheless, the universally rising cost of a post- tem. Using the market analogy in the context of education, secondary degree in America threatens to erode the future the terms firm, seller, school, institution, and university are socioeconomic gains of education. In the same twenty year used interchangeably. Likewise, consumer, buyer, and period as above, average tuition, room, and board rose by student are treated the same. 60 percent after adjustment for inflation, outpacing real Agents in the Q-model prefer to transact with other household income and the availability of financial aid agents in a similar quality range. Coupled with a Classifier (NCES 2008). Furthermore, education grants are increas- System (CS) and Genetic Algorithm (GA) for weighting ingly being displaced in favor of student loans (Baum et al. and creating behavioral rules, respectively, internal agent 2002). The latter trend raises the net cost of education, preferences give rise to an artificially learned equilibrium either pricing students out of pursuing a degree or creating convergence to uniform market structure and stable aggre- debt. Thus, the relationship between price and access to gate behavior. Ortmann and Slobodyan’s (2006) results education is explored here. are also robust to scale. Uncovering the systemic causes and effects of rising tui- However, price is fixed at P = 1 in Ortmann and Slo- tion and student debt on actors in the education system will bodyan (2006) and Vriend (1995), limiting either model’s be vital in developing policies for sustainable market power to simulate rising tuition and debt trends. Due in growth. Awareness of interrelated microeconomic and part to a firm’s quality being a weighted average of current market structural effects of admissions strategies and pric- attending consumer quality Qavg and its own profits π—a ing would likewise benefit an individual institution. To- relationship exactly described later—changing a firm’s 39 price affects firm quality. Changing a firm’s quality shifts the range of consumers a firm prefers to seek. In a recent undergraduate thesis, Drutchas and Érdi (2009) replicate and modify the Q-model to include endogenous price- setting with initial average market price equal to its Ort- mann and Slobodyan (2006) level P*. Average price con- verges to slightly less than P* and aggregate values remain stable. Yet market structure, the distribution of firms across the quality spectrum, exhibits high instability and skewness regardless of profit weight calibrations of a firm’s quality update rule (Drutchas and Érdi 2009). Insta- bility is undesirable from the standpoint of a system mod- eler because it limits replicability of results. High skew- ness of firm quality is undesirable in the market for educa- tion because it limits consumer access. In an effort to move toward a model that has stability, extendability, and explanatory power, the effect of price on market structure in the Q-model is explored systematically. The replication of the Q-model and subsequent experimen- tal results indicate the existence of asymmetric thresholds around P* but beyond which market structure converges to TABLE 3: Summary of Q-model parameters. different equilibria. Model Structure (2) Though some gaps in Ortmann and Slobodyan’s (2006) Unlike consumers in the Q-model, whose initialized description of the Q-model are filled with facts from quality is fixed throughout simulation, firms update their Vriend’s (1995) decentralized economic matching model, quality at the end of each period according to Equation 2 the underlying system structure of the Q-model is repli- (Ortmann and Slobodyan 2006). Ortmann and Slobodyan cated here as closely as possible. Two types of boundedly (2006) calibrate the quality updating parameters α and α rational agents—firms and consumers—interact by seeking 1 2 to fixed values that result in a symmetric steady state transactions that satisfy internal agent preferences and market structure uniformly distributed across the quality making probabilistic stochastic choices. spectrum: All agents are initialized in the first period (semester) with a random quality Q from a uniform distribution to a ‘Symmetric steady state’ denotes situations where quality spectrum M, where Q M [0, 100]. In every time every firm shares the same number of agents, where period agents follow a multi-part economic protocol. every firm serves the same number of agents, where Firms choose both how many units to produce and how every firm has the same profit share . and where the many signals to send to eligible consumers about the units quality of the firm equals the average quality of its produced. In turn, consumers choose to apply to an eligi- consumers. We calibrate the model so that the ble firm from among their received signals. Following average firm quality . equals the average consumer evaluation of consumer applications, acceptance and atten- quality of 50 (which, given our assumption of uniform dance decisions are made. distribution of consumers along the quality spectrum Agents evaluate their success by income earned in the [0,100], is what we can expect on average). current time period. Each agent CS adjusts accordingly the likelihood of repeating their most recent decision if faced Agent Signal Preferences with a similar environment in the future. Consumer Both firms and consumers determine eligibility by check- income I equals a boolean value representing consumer C ing an agent’s quality (QC,F) against their own internal satisfaction: 1 if attending a firm, 0 if not. Firm income IF preferences. Firms randomly choose with replacement a equals the normalized value of profit π as a function of consumer and signal her only if her QC falls within that price P, production Y, demand D, the number of signals S, firm’s range of acceptable consumer quality the marginal production cost C , and the marginal C F Y [QF – Δ , QF + Δ ] (Ortmann and Slobodyan 2006). The signaling cost CS, all scaled by QF (Ortmann and lower bound is the firm’s own preference and allows firms Slobodyan (2006). to signal consumers who are of lesser quality but not be- C yond a low quality tolerance [Q – Δ ]. This reflects the π = [ P · min(Y, D) – CY · Y – CS · S ] · [ QF / 100 ] (1) F tendency of universities to admit applicants of lesser qual- Q = α · Q + α · π (2) t + 1 1 avg 2 ity because they at least generate revenue and at most IF = (π / πavg) · U [0, 1] (3) benefit from education by a higher quality school. A uni- 40 versity would not accept an applicant who would nega- possible actions, but are not subject to a GA and do not tively affect university quality beyond the benefit of mar- change throughout the simulation. As in Ortmann and ginal revenue from that applicant’s attendance. Slobodyan (2006): The upper bound is not a firm’s own preference but in- [Consumers’] behavioral rules each have a conditional stead knowledge of consumers’ preference for only those F part and an action part . The conditional part firms with quality greater than [Q – Δ ]. In other words, a C determines if a rule will be activated . given the consumer will only accept a signal from a firm with Q F current state of the world while the action part greater than a minimum acceptance threshold. Conse- F encodes possible actions. Specifically, rules have the quently, the upper firm preference bound [Q + Δ ] exists F following form: to avoid wasting a signal on an out-of-reach consumer who will only reject a firm’s signal because it is below their minimum acceptable quality.