VOLUME-DEPENDENT MONEY EXCHANGE MODEL FOR GAMING SIMULATIONS Developments in Business Simulation and Experiential Learning, Volume 29, 2002 Volume-Dependent Money Exchange Model for Gaming Simulations Thavikulwat, Precha Towson University pthavikulwat@towson.edu ABSTRACT A central issue of international business is the exchange of money, so a money-exchange model is needed in every international-business simulation. Thavikulwat’s (1999) volume-independent model rewards pedagogically distracting game playing and requires a disconnected fix to avoid meaningless negative valuation under extreme circumstances. A volume-dependent model is proposed that resolves both problems. INTRODUCTION Concurrent with the often-observed globalization of business and the pervasiveness of international-business courses in business-school curricula, college instructors have expressed a need for international-business simulations (Bridwell, 1997). Panel discussions have been presented on international-business simulations (Palia, Yamamura, Cross, Faria, Dickinson, & Roussos,1990; Keys, Edge, & Wolfe, 1992), studies have been conducted on their pedagogical effectiveness (Klein, 1980, 1982), domestically oriented simulations have been adapted to incorporate international content (Halpin & Biggs, 2000), and a scholarly review (Klein, Fleck, & Wolfe, 1993) of international-business simulations is part of the extant literature. Nevertheless, little beyond Thavikulwat’s (1995) overall exposition has been published on the algorithmic requirements of international-business simulations. To be sure, the literature on the algorithmic requirements of business gaming simulations, emerging after Goosen’s (1981) seminal publication, is extensive, a recent review of which has been published by Gold and Pray (2001). Thavikulwat’s recent work on modeling money exchange rates (Thavikulwat, 1999, 2000) does cover the algorithmic requirements of a central international-business issue, but the volume-independent model he presented is problematic in two respects. First, volume independence permits participants to play with the model such as to detract from the pedagogical purpose of the game. Second, in extreme circumstances and in the absence of a disconnected fix, volume independence can give rise to subsequent money values that are negative, therefore meaningless. Accordingly, if the model’s limitation can be avoided by a modification of the algorithm, it should be sensible to do so. This paper explores the problems of Thavikulwat’s volume-independent model and puts forth a modification to the model that allows exchange rates to vary with volume requirements. With the modification, the model gives exchange rates that are the equivalent of dividing each lump sum of money submitted for exchange into infinitesimally small bits, and then exchanging the submitted money for another, bit by bit. As a result, the party demanding the exchange will get the same amount of the other nation’s money, irrespective of how it orders the sum to be exchanged, and the subsequent money value can never be negative. ILLUSTRATION OF THE PROBLEM Thavikulwat (1999) suggested that the relative value of money should be determined by the ratio of the foreign holdings of each nation’s money. Thus, B A BA, F FX = , (1) where XA,B: Value of Nation A’s money with respect to Nation B’s, FA: Foreign holdings of Nation A’s money (e.g., $), FB: Foreign holdings of Nation B’s money (e.g., ₤). Given the model, if a trade transaction requires a trader to exchange one nation’s money for another nation’s, then as Thavikulwat (1999) explained, the subsequent value of Nation A’s money relative to Nation B’s will be: UF TFY + − = B A BA, , (2) where YA,B: Post-exchange value of Nation A’s money with respect to Nation B’s, T: Amount of Nation A’s money given to the trader (e.g., $), U: Amount of Nation B’s money submitted by the trader (e.g., ₤). Now, suppose that foreign holdings of Nation A’s and Nation B’s money amount to A$100 and B₤100, respectively, then the volume-independent exchange rate of Nation A’s money with respect to Nation B’s is equal to their relative value as determined by Equation 1, that is, at A$1 for B₤1. 220 mailto:pthavikulwat@towson.edu Developments in Business Simulation and Experiential Learning, Volume 29, 2002 SOLUTION Suppose further that an importer of Nation B wishes to acquire A$25 for the purpose of paying an exporter of Nation A. Given volume independence, the importer will pay B₤25. The transaction, however, changes the post-event money value, and thus the subsequent exchange rate. Applying Equation 2, it will become A$3 to B₤5, because foreign holdings of Nation A’s money has been reduced by A$25, which went to pay the exporter, and foreign holdings of Nation B’s money has been increased by B$25, which was received from the importer. That is, Both difficulties can be resolved by mathematically fragmenting every transaction such that any amount submitted for exchange is exchanged in infinitesimally small bits integrated over the entire sum, so an importer wishing to sell one nation’s money for another will get the same deal irrespective of whether the amount sold is in one lump sum or several smaller amounts. This approach requires integral calculus, as follows: 5 3 25100 25100 BA, = + − =Y . (3) Let du represent an infinitesimally small increment of the amount of Nation B’s money that is submitted for exchange, let dt represent the infinitesimally small increment of the amount of Nation A’s money that is received in exchange, and let X’A,B be the volume-dependent exchange rate that is desired. Thus, Playing with the model becomes rewarding when the importer wishes to exchange another B₤25 for a second import transaction. This time, the importer will receive only A$15 (i.e., three-fifths of B₤25) in exchange, because of the new money exchange rate. If the importer had combined both exchanges into a single exchange of B₤50 at the start, the importer would have received A$50, which is A$10 more. Thus, volume-independence biases the terms of exchange in favor of those who are able to aggregate many small transactions into a single large transaction. duXdt BA,'= . (6) If the exchange rate is to be volume-dependent, then consistent with the logic of the foreign-holdings model for the importing-exporting case, uF tFX + − = B A BA,' , (7) As for the second problem, the subsequent relative value of money becomes negative when the amount the importer submits for exchange exceeds the foreign holdings of that currency. This occurs because, with volume- independence, where U F FUXT      == B A BA, , (4) t: the cumulative amount of Nation A’s money received in exchange (e.g., $), u: the cumulative amount of Nation B’s money submitted for exchange (e.g. , ₤). Incorporating Equation 7 into Equation 6, we have which when incorporating into Equation 2 gives du uF tFdt + − = B A , (8) ( ) ( )UFF UFFY + − = BB BA BA, . (5) which can be rearranged as follows: Thus, when U > FB, YA,B becomes negative. uF du tF dt + = − BA . (9) Of course, the circumstance that can cause the post- exchange money value to turn negative is an extreme one, and the fault can be fixed whenever it would occur by increasing foreign holdings of both nations’ money by an equal and sufficiently large amount. Nevertheless, the fix would be disconnected from the basic model itself. It need not be accepted, because a more elegant solution is possible. If, as before, U represents the entire sum submitted for exchange and T represents the entire sum received in exchange, then , (10) duU Uu u∫ = = = 0 221 Developments in Business Simulation and Experiential Learning, Volume 29, 2002 , (11) dtT Tt t∫ = = = 0 If instead, the importer splits the B₤50 into two B₤25 amounts, the importer will receive A$20 for the first exchange, computed as follows: and from Equation 9, 00.20 25100 25100 1 = + × =T . (20) ∫∫ = = = = + = − Uu u Tt t uF du tF dt 0 B0 A , (12) which becomes ( ) ( )uFtF Uu u Tt t +=−− = = = = B 0 A 0 lnln , (13) For the second exchange, the foreign holdings of both nations’ money must first be adjusted for the result of the first exchange, which increased FB by B₤25 and reduced FA by A$20. So, the importer will receive A$13.33 for the second exchange, computed as follows: ( ) ( ) 33.13 2525100 2520100 2 = ++ ×− =T . (21) and reduces to       + =      − − B B A A lnln F UF F TF . (14) Thus, for both exchanges, the importer will receive A$33.33, the sum of T1 and T2. As the amount received is the same irrespective of whether the importer chooses to exchange the money in one lump sum or two separate amounts, the model does not produce a windfall gain for participants under any circumstance. Thus,       + =      − B B A A lnln F UF TF F , (15) This additive characteristic of the model can be proved mathematically for all combinations of U1 and U2 that add to U. If a trade-enabling money-exchange transaction of size U is split into two parts, U1 and U2, then following Equation 17, the receipt from the first transaction (T1) is and 1B 1A 1 UF UF T + = . (22) B B A A F UF TF F + = − . (16) The receipt from the subsequent second transaction (T2) is After collecting term, Equation 16 becomes UF UF T + = B A . (17) ( ) ( ) 21A 21A 2 UUF UTF T ++ − = . (23) Summing both transactions and rearranging terms gives Thus, the effective exchange rate (X”A,B) for an import- export transaction is: ( )       ++ −+ ++ =+ 21B 2 1 21B 2A 21 1 UUF U T UUF UF TT . (24) UF F U TX + == B A BA," . (18) Incorporating Equation 22 into Equation 24 results in ( ) ( ) ( )21B 21A 21B 2 1B 1A 21B 2A 21 1 UUF UUF UUF U UF UF UUF UF TT ++ + =      ++ − + + ++ =+ . (25) For the example problem given earlier, if the importer submits at the start B₤50 for money exchange, the importer will receive A$33.33, applying Equation 17 as follows: 33.33 50100 50100 = + × =T . (19) Thus, because UUU =+ 21 , (26) 222 Developments in Business Simulation and Experiential Learning, Volume 29, 2002 223 it follows that , (27) TTT =+ 21 Goosen, K. R. (1981). A generalized algorithm for designing and developing business simulations. Developments In Business Simulation & Experiential Exercises, 8, 41-47. proving that the amount received in exchange will be the same irrespective of whether the amount submitted for exchange is submitted in one lump sum or in any two smaller amounts. Generalizing from this result, it follows that the amount received in exchange will be the same irrespective of how the sum may be split for the exchange. Halpin, A. L., & Biggs, W. D. (2000). Internationalizing the introduction to business course using an international text and a domestic simulation with a twist. Developments In Business Simulation & Experiential Learning, 27, 115-121. Finally, the post-exchange money value can be gotten by incorporating Equation 17 into Equation 2, giving ( )2 B BA B B A A BA, UF FF UF UF UFF Y + = + + − = . (28) Keys, J. B., Edge, A., & Wolfe, J. (1992). Using simulations to teach international issues: An analysis of the multinational management game’s learning environment. Developments In Business Simulation & Experiential Exercises, 19, 245. Klein, R. D. (1980). Can business games effectively teach business concepts? Developments In Business Simulation & Experiential Exercises, 7, 128-131. Klein, R. D. (1982). Problems associated with the assessment of experiential learning using the multiple choice test. Developments In Business Simulation & Experiential Exercises, 9, 221-223. Clearly, the post-exchange rate cannot be negative because none of the terms are negative. The post-exchange negative valuation problem is therefore also resolved. Klein, R.D., Fleck, R.A., & Wolfe, J. (1993). A role for management games in internationalizing the business school curriculum. Journal of Management Education, 17, 159-173. CONCLUSION Palia, A. P., Yamamura, H., Cross, L., Faria, A. J., Dickinson, J. R., & Roussos, D. S. (1990). International management simulation gaming: Current status and future developments. Developments In Business Simulation & Experiential Exercises, 17, 229. The problems of a volume-independent money exchange model are solvable by a modification of the basic model to make it volume dependent, as presented above. Granted, the circumstances giving rise to the problems of game playing and negative valuation may be extreme, but extreme circumstances are not necessarily unlikely when models are used in gaming simulations. Gaming simulations are themselves extreme simplifications of reality. It is not their reality, but their irreality that makes them pedagogically valuable. Thavikulwat, P. (1995). Computer-assisted gaming of international business. Developments In Business Simulation & Experiential Learning, 22, 67-73. Thavikulwat, P. (1999). A model of currency exchange rates. Developments In Business Simulation & Experiential Learning, 26, 200-208. The real world is cluttered with irrelevant events that detract from learning and confound assessment. Gaming simulations are useful because they reduce the clutter. The world of gaming simulation is a world of very few people and very few nations. It thus is entirely conceivable that one of these very few people might control most of the money that flows among the nations. Models used to exchange money under these circumstances must therefore be especially robust. Thavikulwat, P. (2000). Validating a model of currency valuation. Developments In Business Simulation & Experiential Learning, 27, 132-137. REFERENCES Bridwell, L. (1997). College students need simple computer simulations, especially for international business courses. Developments In Business Simulation & Experiential Learning, 24, 195-196. Gold, S. C., & Pray, T. F. (2001). Historical review of algorithm development for computerized business simulations. Simulation & Gaming, 32, 66-84. Table of Contents Volume 29, 2001 Threshold Marketer A Family Of Marketing Simulations: Basic Marketer And Advanced Marketer Team Mode And Solo Mode Globalization As An Extended Experiential Exercise The Benefits And Planning Considerations Of Short Term Study Abroad Programs How 2 Setup Your Office Computer To Run Linux For Teaching E-Commerce Without Messing Everything Else Up Incorporating Cosmopolitan-Related Focus-Group Research Into Global Advertising Simulations Demonstration Of Advanced Features In Computer-Assisted Gaming Of International Business A Comparison Of Discrimination-Based Versus Conventional Simulation Game Scoring A Universal Mathematical Law Criterion For Algorithmic Validity The Impact Of Public Policy On Innovation: A Simulation Project For Research And Teaching Participant Identification Of Competitors In A Marketing Simulation Competition Simulation Research In The Hospitality Industry "Computer Simulation, Games And Roleplay: Drawing Lines Of Demarcation" Simulation Distribution Alternatives: Author/User Considerations Managing The Curiosity Gap Does Matter: What Do We Need To Do About It? Use Of External Interventions In A Computer Based Simulation Putting Service Learning Into Orbit It's A Wonderful Life: Simulating The Golden Years Adventures In Creating An Outdoor Leadership Challenge Course For An Emba Program Vbotz: A Pedagogical Cross-Disciplinary, Multi-Academic Level Manufacturing Corporate Simulation Use Of Computer Modeling In Management Accounting Is Simulation Performance Related To Application? An Exploratory Study Learning Cooperatively May Not Be Learning Collaborately! Perception Is Reality: Sharing Frames International Management Virtual Teamwork: A Simulation Financial Plan For Your Life And Career Goals Using Project-Based Experiential Learning Groups In The Principles Of Marketing Course Futures Course: Learning How To Anticipate The Future Of Business Interactive Online Strategic Market Planning With The Web-Based Boston Consulting Group (BCG) Matrix Graphics Package Strategy Learning In A Total Enterprise Simulation Investigation Of The Impact Of Decision Parameters For A Dutch Auction Simulation For Ipo Issues Integrating In-Class Learning With Out-Of-Classroom Experiences Through A Managerial Competency Development Framework War And Peace: Managing Students Learning Experience In A Competitive Simulation Game Virtually Experiential Classrooms Exercise: Conducting Role Plays Using Student Generated Cases Procedural Justice And Acceptance In Group Decision Making The E-Commerce Game: A Strategic Business Board Game Does Student Preparation Matter In A Simulation? A Comparison Of Pedagogical Styles The Game Of Business - A Weekend MBA Course Volume-Dependent Money Exchange Model For Gaming Simulations Implementing Service Learning For Accountants: The Not For Profit Project To Teach Vikings To Behave Among Mandarins: Lessons From Teaching With A Simulation Model Of Applied Business Ethics In International Management Maze Bright Teachers In The Classroom The Validity Investigation Of A Test Assessing Total Enterprise Simulation Learning What Makes Strategy Possible: An Illustration Using Paper And Scissors Learning Micro-OB Skills While Making Top Management Decisions In A Multinational Industrial Firm The Absel Research Heritage And The Bkl: Leveraging Their Value For Future Research New Product Development (Npd) Simulations: Some Challenging Questions And Tough Modeling Issues The Biofeedback Stress Test The Power Circle Exercise Total Enterprise Simulation Learning Compared To Traditional Learning In The Business Policy Course A Business Game Distance Education Application: Learning Outcomes And Experiences Is the Tobin's Q a good Indicator of a Company's Performance? A Critical Examination of the "Experiential" Premise Underlying BUsines Simulation Usage