Abstract
Background. Although fairness is central to society and to games that are taken seriously, the
Aim. To resolve the problem of structural fairness in
Method. Mathematics and examples are used to clarify positions, present proofs, and show application.
Argument. Structural fairness is of three kinds: positional, order, and arrival.
Finding. For fixed number of parties,
Application. The rotational procedures apply to business games with modeled and real markets, and may apply to all games with a scoring system that is taken seriously.
Conclusion. Games can be structurally fair, but the game that is structurally fair must be a multi-episodic game that incorporates fairness into its design. For assuring structural fairness, proportional and random methods are generally inferior to rotation.
Keywords
Introduction
Consider the hypothetical example of a classically designed business game in which all firms require the same opportunities (or resources) from a single modeled entity. The firms are each managed by a team of students, who submit decisions for processing on an episode-by-episode basis, a classical business-game design pioneered in the early 1950s (Wolfe, 1993) and still in common use today. Suppose the opportunities required are labeled building permits and the modeled entity is said to be the government. In this case, the game would simulate the building construction industry. For this business game, and the many other games with a limited-opportunity element, the question of interest to game designers is how building permits (or opportunities) should be allocated to firms if the number of building permits is fewer than the total number sought by the firms.
The question of how limited opportunities should be allocated is the fundamental question in the study of economics, which generally prefers that the allocation should be based on free-market processes. In the case of building permits, the requirements of the firms might be addressed by asking the firms to bid for available permits. Yet, if a free-market process is not built into the game, then what allocation rule should apply? Moreover, if a free-market process is used, is fairness necessarily assured?
In fact, the use of free-market processes wherein players trade with other players (Cannon, Yaprak, & Mokra, 1999; Thavikulwat, 1997) is relatively new to business games. The vast majority of business games in common use depend on mathematics to model the market (Cannon, Cannon, & Schwaiger, 2009; Cannon & Schwaiger, 2005; Gold & Pray, 2001; Goosen, 2009; Teach, 2007; Wolfe & Gold, 2007). Of the various models, the classic one is Gold and Pray’s (1983, 1984, 1990) (GP), which models a market of sales opportunities in three-steps. First, compute the number of opportunities (e.g., quantity demanded by the market) at the industry level (e.g., for all firms combined). Second, allocate the opportunities to the firms. Third, reallocate opportunities in excess of product availability (e.g., stock outs) to the firms with a shortage from the second step, but only when stock outs are extreme, a condition defined by (a) market share that is more than three standard deviations from the mean of all firms in the same episode and (b) demand exceeding “possibly . . . between two or three” (Gold & Pray, 1990, p. 128) times the number of units available for sale. The first two steps allocate opportunities without regard to product availability. The third step accounts for product availability, but only in the event of an extreme stock out. In that event, Gold and Pray (1990, pp. 133-135) suggests that the excess demand should be reallocated proportionally to firms with remaining stock based on new market-share computations across firms that have remaining stock. Otherwise, the stock out is disregarded.
Substituting the more general term requirements for product availability, the three-step approach may be said to be insensitive to requirements. Requirements-insensitive allocation may be satisfactory for markets where opportunities are truly independent of requirements, so that fairness is not an issue. In the general case where opportunities depend upon requirements, however, fairness could be an issue.
To be clear, stock outs, whatever their causes, are by themselves not a fairness issue. The reallocation of product demand from firms with stock outs to other firms, however, could be a fairness issue. Thus, a child at a party who refuses a cookie is not a fairness issue, but the reallocation of that cookie to the other children could be a fairness issue.
When fairness is an issue, fairness must rule, for as Rawls (1957, 2001) has argued, without fairness there can be no justice; without justice, there can be no well-ordered society. Fairness may be especially important in a game, for although a game is an “activity standing quite consciously outside ‘ordinary’ life as being ‘not serious’” (Huizinga, 1950, p. 13), games are supposed to be fair, so the players may have less tolerance for unfairness in a game than they would have in everyday-world activities.
Even so, the opposing position should be considered. If the game is structurally unfair, will it make a difference in the outcome and will the participants notice? Perhaps not. A study by Wolfe and Jackson (1989) found that a glitch deliberately introduced into The Business Management Laboratory’s (1983) algorithm “had no discernible effect on the player’s perceptions of the game’s realism, nor on each team’s economic performance” (p. 284). One might protest that their finding applies narrowly, because anyone desiring to place a glitch that has a discernible effect on perception and performance can surely do so. On the other hand, one also might affirm, as Wolfe (1991) did, that “because a game’s algorithms are only one part of a viable experiential learning environment, greater emphasis should be placed on how players cognitively and emotionally interact with the model” (p. 362). Yet, greater emphasis of one should not mean neglect of the other, for the argument for fairness is fundamentally an ethical one. If a game that is taken seriously can be made fairer, it should be made fairer, for fairer is better, and better is right.
What Is Fairness?
Fairness, formally defined, is equality of opportunity, a philosophical definition understood to represent a political ideal in a society of necessarily unequal opportunities (Stanford Encyclopedia of Philosophy, 2015). The definition begs the question of what constitutes an opportunity in a game of competing parties, where a party may be an individual, team, or community, depending upon the particulars of the game. If the opportunity is to be first—first to make a move, as in chess, or first to receive building permits, as in the supposed building-construction game—then only one party in a game can be first. If the toss of a coin should decide who is first, then that procedure may be acceptable to the players, but the fairness of the procedure depends on what happens later, for when things happen matters. Thus, fairness is about sequence, which together with synchronization and frequency constitutes the three elements of Moore’s (1963) sociological analysis of time. If you hit me, that may not be fair, but if I hit you in return, then that should be fair.
More generally, fairness has two aspects: a structural aspect and a behavioral aspect. Structural fairness means that the architecture of the game is proper. Yet, even when the game is structurally fair, players may cheat, the behavioral aspect. This investigation, confined to structural fairness, recognizes but does not address cheating.
Structural fairness is especially relevant for transaction-based games, which excludes classically designed business games where, as Teach (1990) has observed, “simulation players never develop the concept that company sales are the result of many individual transactions” (p. 115), because their sales are derived by formulae that aggregate and allocate modeled-market demand. In fact, the recipes discussed herein were developed for a real-market game that is necessarily transaction based, but in the interest of addressing the familiar before the unusual, the discussion of how recipes for structural fairness apply in a limited way to classical games precedes the discussion of how the recipes apply more expansively to a real-market game.
Structural fairness may be assured by rotating the sequence of play between competing parties. In a game of two parties, rotation alternates the first-mover (or last-mover) advantage between the two, which requires the game to have an even number of episodes. In this case, the recipe is simple. If you make the first move in the first episode, I make the first move in the second episode, and so forth. For the building-construction game, if your building-permit requirements are satisfied first in the first episode, with permits remaining left for me; then my building-permit requirements should be satisfied first in the second episode, with permits remaining left for you.
The recipe for a game of many competing parties, however, is not so simple, for the number of ways many parties can be sequenced for rotation becomes rapidly very large as the number of parties rise, and only a subset of those ways are optimal. Thus, the number of ways by which a 6-party game can be sequenced is 6! = 720, too many to consider rotating through all of them in a single gaming event. The problem then is to identify the subset that is optimal for a realistic number of episodes in a single gaming event. As will be shown, the optimal subset for complete fairness in a 6-party game consists of only 6 episodes. More generally, any N-party game requires no more than 2N episodes for complete fairness when N is odd and no more than N episodes when N is even.
Kinds of Structural Fairness
To arrive first at a common understanding of terminology, I shall begin with a comprehensive exposition on structural fairness. In my view, a game of many parties and many episodes may require three kinds of structural fairness: positional fairness, order fairness, and arrival fairness.
Positional fairness is of concern when every party requires as much of the contested item as any other party. In this case, a party’s position in the sequence is important but the identity of the party ahead is unimportant, because a difference in identity will not give rise to a difference in opportunity. Thus, if every theatre goer can purchase only one ticket and the theatre does not have enough tickets for everyone, then one’s position in the ticket line matters, but the desire for tickets by those ahead of the line does not matter, because however many tickets each of those in front desire, each can purchase only one.
Order fairness is of concern when some party requires much more of the contested items than other parties do. In this case, standing behind the high-requirements party is especially disadvantageous irrespective of one’s position in the line. Thus, if every theatre goer can purchase as many tickets as desired until all tickets are sold, being second in line behind the scalper who will buy all remaining tickets is worse than being third in line but ahead of the scalper.
Arrival fairness is of concern when different parties arrive at the distribution point at arbitrarily different times. In this case, giving early arrivals the advantage of earlier service would be capricious. Thus, if bulletins are periodically posted stating that a limited number of desirable tickets are available, first-come-first-served, then those who pass more frequently by the bulletin board are capriciously advantaged over others.
All three kinds of fairness can be achieved by rotation over a sufficiently large number of episodes; the objective is to maximize fairness over the fewest number of episodes. The exposition that follows expands on each kind of fairness and explains how each is optimally addressed by a specific rotational procedure, or recipe. The exposition is technical, so those interested only in a conceptual understanding of how fairness might apply in a business game may skip the rest of this section and go directly to next section, Alternatives to Rotation.
To minimize tedium, the exposition that follows assumes that the number of positions equals the number of parties. The assumption preserves generality, because dummy positions or dummy parties can be added for a perfect match whenever the number of positions do not equal the number of parties.
Positional Fairness
I define complete positional fairness to mean that every party occupies each position in a sequence as frequently as any other party does. The necessary condition for complete positional fairness is that the number of episodes must be an integer multiple of the number of parties. In a six-party, six-episode game, complete positional fairness is achieved by rotating the assignment of parties between episodes, as illustrated in Table 1, where the six parties are identified by the letters A through F. The rotation proceeds as follows:
Assign the parties by a convenient process, such as alphabetical order or drawing lots, to all the positions of the first episode.
Let xi,j refer to the party assigned to episode i and position j. Assign ordered letter to the parties of the first episode, A to x1,1, B to x1,2, C to x1,3, and so on up to N, the number of parties, so the number of positions equals the number of parties.
Transpose the assignments of the first episode to the first position of N episodes, so the number of episodes also equals the number of parties.
Then for i = 2 through i = N and j = 2 through j = N, assign xi, j = xi,j−1 + 1, where xi,j−1 + 1 refers to the next letter after xi,j−1, wrapping from the Nth letter back to the first letter when that next letter would exceed the Nth party.
Complete Positional Fairness to Six Parties (A Through F) Over Six Episodes.
This positional rotation procedure has an additive character that can been seen clearly if the parties are assigned integers, conveniently starting with zero (0), rather than letters such that 0 replaces A, 1 replaces B, 2 replaces C, and so on. Then the procedure reduces to Equation 1.
The result is a Latin-square assignment that may be duplicated exactly in sets of Ns if additional episodes are desired. Equation 1 can be simplified by dropping the offsetting constant, in which case the formula becomes Equation 2. The effect of dropping the constant is to rotate all sequences by two positions, without losing positional fairness.
Order Fairness
I define complete order fairness to mean that the relative place of a party to every other party occurs with the same frequency for all parties, where relative place refers to a party either preceding or following another party. In the case of the six-party, six-episode game arranged as shown in Table 1, A precedes B five time, in the first and third through sixth episodes, whereas B precedes A only once, in the second episode, so order fairness is incomplete.
In the case of the four-party, four-episode game arranged as shown in Table 2, order fairness is complete. To see this, note that the number of ordered pairs for any number (N) of parties is equal to N × (N – 1), so for 4 parties we have 4 × 3 = 12 ordered pairs, namely, AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, and DC. The AB order occurs in episodes 1 (A-B) and 3 (A-D-B), the AC order occurs in episodes 1 (A-B-C) and 2 (A-C), and so forth. Every ordered pair occurs twice. Similarly, order fairness is complete in the six-party, six-episode game arranged as shown in Table 3, for every ordered pair occurs thrice.
Complete Positional and Order Fairness to Four Parties (A-D) Over Four Episodes.
Complete Positional and Order Fairness to Six Parties (A-F) Over Six Episodes.
Positional fairness also is complete in both of these arrangements, as inspection verifies. To wit, consider any party, such as B. Notice that B appears once and only once in every position of both Tables 2 and 3. The same is true for any other party, proving complete position fairness.
The arrangements of Tables 2 and 3 were generated by adjusting rotational assignments. The procedure proceeds as follows:
Take the first three steps of positional rotation.
Add a dummy N + 1 party.
Then for i = 2 through i = N and j = 2 through j = N, assign xi, j = xi,j−1 + i, where xi,j−1 + i refers to the ith letter after xi,j−1, wrapping from the (N + 1)th letter back to the first letter when that next letter would exceed the (N + 1)th party.
This order rotation procedure has a multiplicative character that can be seen clearly if the parties again are assigned integers rather than letters such that, as before, 0 replaces A, 1 replaces B, 2 replaces C, and so on. Then order rotation for i < N + 1 reduces to:
More generally, order rotation for any i is:
Where
Order rotation assures complete positional and order fairness under two conditions: (a) the numbers of episodes is an integer multiple of the number of parties and (b) N + 1 is a prime number. This assertion can be proved. To simplify the proof without losing generality, assume i < N + 1, enabling the proof to be based on Equation 3.
To prove positional fairness, consider Table 4, derived from ij − 1, Equation 3 before the modulus. Notice that the items in every row of Table 3 are transposed into every column, thus the items of row 3 (2, 5, 8…) are the same as the items of column 3 (2, 5, 8…). This is so because swapping i and j in ij – 1 leaves the results unchanged. Since every row is transposed into a column, proving that the N consecutive items of every row computed from Equation 3 are unique suffices to prove positional fairness.
Order Rotation Before the Modulus.
If a row of Equation 3 should contain two items that are identical, then the difference between the dividends of both items must be (ij2 − 1) − (ij1 − 1) = i(j2 − j1) = k(N + 1), where j1 and j2 refer to the positions of the two items and k can be any integer. The second equality is required for the modulus to yield the same item but the second equality is impossible when N + 1 is a prime number, which cannot be factored, because both i and j are less than N + 1.
Furthermore, the set of positive integers that can be residuals of any positive integer mod (N + 1) is bounded by 0 and N, but the items (xi,j) themselves are bounded by 0 and N – 1, so N must not fall within the bounds of i < (N + 1) and j < (N + 1). In fact, N lies just outside of the bounds, as the residual of the modulus (N + 1) when i = (N + 1) or j = (N + 1) or both. Thus, the N consecutive items of every position of an episode must be unique and bounded by 0 and N – 1, proving positional fairness.
To prove order fairness, consider Table 5, derived from Equation 3 after applying modulus N + 1 = 7, thus N = 6. Notice that the items are inverted in positions 1 and 6, positions 2 and 5, and positions 3 and 4. Likewise, the items are inverted in episodes 1 and 6, episodes 2 and 5, and episodes 3 and 4. The inversions are specific to N = 6.
Order Rotation After Modulo 7.
The inversions mean that the dividend of the inverted items differ by k(N + 1), where k, as before, can be any integer. The inverse of episode i is episode N + 1 – i = iʺ and the inverse of position j is N + 1 – j = jʺ, so the dividend-differences for inversion is as follows:
Equation 6 is true because its left side reduces to (N – i – j)(N + 1), so every position truly is inverted in another position. This implies that whenever an item precedes another item in an episode, that item follows the other item in another episode, proving order fairness for any N items.
Accordingly, order rotation assures both complete positional fairness and complete order fairness when the number of episodes is an integer multiple of the number of parties and the number of parties is one less than any prime number, which covers Ns of 2, 4, 6, 10, and 12. By adding a step, both N = 3 and N = 8 also can be covered.
For N = 3, complete positional and order fairness can be achieved for episodes that are multiples of six by stacking two 3 × 3 Latin squares created by positional rotation such that the order of the parties of every episode is reversed between the two Latin squares. The result is a set of six sequences composed of all 3! = 6 possible sequences of 3 parties, so both positional and order fairness are assured.
Table 6 shows two stacked 3 × 3 Latin squares so constructed. The formula for the second N × N Latin square of the stack, i > N, is given in Equation 7. As with Equations 1 and 2, Equation 7 also can be simplified by dropping the offsetting N − 1, in which case the formula becomes Equation 8.
Stacked Arithmetic Rotation for Complete Positional and Order Fairness to Three Parties Over Six Episodes.
The stacking procedure extends to Ns of any size. Stacking assures positional fairness because the first N × N Latin square is constructed by positional rotation and the second N × N Latin square merely inverts the order of the positions. Thus, in Table 6, the parties in the first position of the second Latin square are the parties in the third position of the first Latin square, and vice versa. Stacking also assures order fairness because each sequence of the second N × N Latin square is constructed by reversing the corresponding sequence of the first Latin square. So, complete positional fairness and complete order fairness is obtained in any N-party game in 2N episodes by positional rotation over the first N episodes and reversing the sequential ordering of the parties over the second N episodes.
For N = 8, eight is twice four, so an 8 × 8 Latin square can be constructed by tiling two 4 × 4 Latin squares on an alternating basis, each 4 × 4 Latin square constructed by including only four parties excluded from the other 4 × 4 Latin square. Table 7 shows the result of tiling that starts with the 4 × 4 Latin square of Table 2. Accordingly, for an eight-party game, tiling gives rise to complete positional fairness and complete order fairness in eight episodes, less than half the number of episodes that stacking requires. The tiling procedure applies whenever N is even, for the diagonal tiles can be composed of stacked N/2 × N/2 Latin squares should N/2 be other than one less than a prime number.
Tiled Order Rotation for Complete Positional and Order Fairness to Eight Parties Over Eight Episodes.
Other cases that avoid stacking generally require a choice between complete positional fairness and complete order fairness. Positional rotation assures complete positional fairness whenever the number of episodes is an integer multiple of the number of parties, because rotation causes each position to be occupied by the next party in the next episode. Order rotation by adding dummy parties, and positions, until the number of parties is one less than the next prime number assures complete order fairness when the number of episodes also is an integer multiple of the number of parties, dummies included. Applying this last procedure to seven parties using Equation 3 and modulo 11 gives rise to Table 8, where the dummy parties appear as blanks in the table. Tightening the table by shifting parties to the left to occupy blanks, essentially skipping over dummies, gives rise to Table 9. Notice that the order of the parties in Table 9 is reversed between episodes 1 and 10, 2 and 9, 3 and 8, 4 and 7, and 5 and 6, so order fairness is complete.
Order Rotation for 7 Parties Over 10 Episodes Using Modulo 11 and Showing Dummy Parties as Blanks.
Order Rotation for 7 Parties Over 10 Episodes Using Modulo 11 and Shifting Parties to Occupy Blanks.
Arrival Fairness
I define complete arrival fairness to mean that every party added to a sequence is inserted into a position with the same frequency as any other position in the sequence. For example, if party D is added to the X1-X2-X3 sequence, where each X represents a party of the existing sequence, D should appear in the first through fourth position with equal frequency. Thus, the sequence of four parties over four continuous episodes that yields complete arrival fairness could be D-X1-X2-X3, X1-D-X2-X3, X1-X2-D-X3, and X1-X2-X3-D, but it would not be D-X1-X2-X3, D-X1-X2-X3, X1-X2-D-X, and X1-X2-X3-D. In the former case, party D appears once in the first through fourth position over the four episodes; in the latter case, party D appears twice in the first position and does not appear at all in the second position over the four episodes.
To construct the arrival-fair sequence for each episode, my formula, applied successively to each party from the first party (n = 1) to the last party (n = N) as that party is added to the sequence, is as given in Equation 9.
For example, if the first episode (i = 1) involves three parties, the first party (n = 1) is assigned to position 1 − (1 mod 1) = 1 − 0 = 1. The second party (n = 2) is assigned to position 2 − (1 mod 2) = 2 − 1 = 1, which pushes the first party to the second position, so the two-party sequence is 10, or BA, where, as before, A = 0 and B = 1. The third party (n = 3) is assigned to position 3 − (1 mod 3) = 3 − 1 = 2, which pushes the first party from the second position to the third position, so the three-party sequence is 120, or BCA. Applying this procedure to six episodes gives rise to the set of sequences shown in Table 10.
Arrival Rotation for Three Parties Over Six Episodes.
Arrival rotation applies particularly to multi-player, transaction-based games when players cycle through the game in waves, so that the first-come-first-served rule would be an unsuitable basis for adding players to a sequence constructed from the earlier wave, such as a multi-player game jointly played by classes that meet at different times. In this case, if first-come-first-served is used to sequence players for contested items, classes meeting earlier would be advantaged over classes meeting later, because the players of the earlier classes would, ceteris paribus, generally submit their decisions sooner and therefore be ahead in the queue than players of the later classes.
Alternatives to Rotation
Proportional allocation, wherein opportunities are allocated in proportion to requirements, and random allocation may appear to be simpler means of assuring fairness than rotation. The problem with these two methods, however, is that they each require conditions that are more restrictive.
Proportional Allocation
Proportional allocation fails when opportunities cannot be allocated exactly in the proportions desired. To see why, consider again the building-construction game of the introduction. Suppose in every episode of the game, 10 building permits are available when the six firms of the game require a total of 21 building permits, the requirements distributed such that Firm A requires 1 permit; Firm B, 2 permits; Firm C, 3 permits; and so forth, as shown in Table 11. The objective is to allocate fairly the building permits to the firms over six episodes, given that the firms are the same in all other respects. This objective means that at the end of six episodes, the firms should have as many building permits in proportion to the total the number of building permits issued, as the firms require in proportion to the sum of the requirements of all firms. Thus, the proportion of permits distributed across the six firms after six episodes should be as close as possible to the proportion of requirements shown in Table 11.
Distribution of Permits Required and Available.
Table 12 shows the distribution of permits resulting from positional rotation, applying the sequences of Table 1. In this case, for episode 1, the distribution begins with Firm A, which gets 1 permit; followed by Firm B, 2 permits; Firm C, 3 permits; and Firm D, 4 permits. The 10 available permits having been distributed, no permit remains for Firm E and Firm F. In episode 2, the distribution begins with Firm B, which gets 2 permits; followed by Firm C, 4 permits; and so forth as shown in Table 12.
Distribution of Permits by Positional Rotation.
Table 13 shows the distribution of permits resulting from order rotation, applying the sequences of Table 3. Table 14 shows the distribution of permits resulting from arrival rotation, extending the sequences of Table 10 from three parties over three episodes to six parties over six episodes. After 6 episodes, 60 permits will have been distributed in the proportions shown in Tables 12 through 14 for the three rotational methods.
Distribution of Permits by Order Rotation.
Distribution of Permits by Arrival Rotation.
As to proportional allocation, allocating the 10 building permits in proportion to requirements means that the permits should be allocated as shown in the last two rows of Table 11, depending on the divisibility of permits. Considering that building permits should be indivisible, the allocation could be adjusted to give Firm A one permit instead of zero, so that the sum of all permits proportionally allocated equals the 10 available. The adjustment resolves a problem of proportional allocation: When the opportunities are in indivisible units, the sum of opportunities allocated may not equal the number of opportunities available. The result of applying adjusted proportional allocation to six episodes is shown in Table 15.
Distribution of Permits by Adjusted Proportional Allocation.
The root-mean-squared difference between the proportion of permits required (Table 11) and the proportion of total permits distributed over six episodes applying positional rotation (Table 12), order rotation (Table 13), arrival rotation (Table 14), and adjusted proportional allocation (Table 15) is shown in Table 16. Order rotation gives rise to a difference of .007 that is less than a quarter of the difference of the next smallest difference of .033, the result of adjusted proportional allocation. A graph of the differences is shown in Figure 1. Clearly, order rotation is the method that most closely matches requirements, so it is the fairest of the four allocation methods for the building-construction game. The superior performance of order rotation over proportional allocation does not extend to all cases of indivisible limited opportunities, but the fact that order rotation can sometimes be better should deter game designer from uncritically relying on proportional allocation.
Root-Mean-Squared Difference of Allocation Methods.
Proportional differences of allocation methods by required building permits.
Random Allocation
Random allocation equalizes expected opportunity, not realized opportunity. True fairness requires that opportunities be realized, not merely expected. Yet, one might argue that if a game is administered for many episodes beyond the 4 to 12 that is typical (Anderson & Lawton, 1992; Rollier, 1992), the law of large numbers will assure a fair outcome with random allocation. The argument is fallacious, because large numbers do not eliminate runs. Runs are why, as Styer (2000) has observed, stars are not uniformly distributed in the sky even though stars are born of a random process. Moreover, the relative performance of competing parties tends to be correlated across the episodes of business games, which explains why early dominance has been shown to be a notable problem (Bernard & de Souza, 2009; Patz, 1992, 1999, 2000; Peach & Platt, 2000; Rollier, 1992; Teach & Patel, 2007), even if the problem is not always found (Wolfe, Biggs, & Gold, 2013). Random selection gives rise to runs that accentuate an early advantage. Rotation forestalls runs, ameliorating advantage.
In fact, random allocation is truly fair only for the relative positions of parties in the first episode. In this restrictive circumstance, random allocation may forestall cheating that requires knowledge of positions before the competitive event begins.
Modifying the Gold and Pray Model
The GP model is based on a log-linear demand function (Gold & Pray, 1990, p. 123) that assumes that demand is independent of supply, which is to say, more generally, that opportunities are independent of requirements. The demand-supply independence assumption may be more constraining than it appears, considering that suppliers with more for sale tend also to be advantaged by being more visible, being more likely to have sufficient stock to satisfy a large order, and being seen as a less risky trading partner, because the large-volume supplier has more to lose by mistreating a customer.
Thus, if opportunities should depend upon requirements, the question arises as how the requirements-insensitive GP model might be modified to include requirements more directly within the model, ergo, to allow demand to depend on supply. One way to do this is to add supply as an independent variable of the demand function. In this case, Gold and Pray’s (1990, p. 123) log-linear demand function becomes Equation 10, where Q is the quantity demanded; P, price; M, marketing expenditure; R, research and development expenditure; S, supply; and a, b, c, e, and g are parameters. This method treats supply as simply another independent variable, without recognizing supply’s unique role in sales.
Another way is to keep the original demand function, but to substitute order rotation for the third step, when stock outs are reallocated. Thus, the sum of quantities demanded in excess of quantities available would be distributed by order rotation to firms whenever quantities available exceed quantities demanded, forgoing Gold and Pray’s (1990, p. 135) checks for extreme values. This second method ties demand to stock outs, consistent with the view that customers preferentially shop at firms with high supply because high-supply firms are less likely to have stock outs.
For example, consider a game of three competing service firms, all of which submit exactly the same marketing decisions for every episode of a six-episode game. The difference between the three firms is operational. For every episode, Firms A, B, and C supply 1, 7, and 9 service units, respectively, as shown in Table 17. The everyday-life scenario that corresponds to the game could be three barbershops in the same shopping center, the first having one chair, the second having seven chairs, and the third having nine chairs.
Allocation of Stock-Out Demand in One Episode.
For this example, the GP model computes the same gross demand quantity, say 5 units, for every firm in every episode, as shown in Table 17. The 5-unit gross demand of Firm A exceeds its one-unit supply, so the excess of 4 units is allocated to the other two firms. The GP model allocates by gross demand, which is the same 5 units for Firms B and C. Accordingly, Firms B and C are each allocated 2 units, and net demand is 1 for Firm A and 7 each for Firms B and C, as shown on Table 17.
To allocate stock outs by order rotation, the 4 excess units can be allocated B-C or C-B, as shown in Table 17. For B-C, 2 units go to Firm B to match the 7-unit supply; the remaining 2 units go to Firm C. For C-B, 4 units go to Firm C to match the 9-unit supply, leaving nothing for Firm B. For complete order fairness over six episodes, B-C and C-B allocations must each occur three times. The root-mean-squared difference between the supply and net demand after six episodes is larger for GP net demand than for order-rotated net demand, as Table 18 shows, so order rotation is better.
Root-Mean-Squared Difference Between Supply and Net Demand Over Six Episodes.
In this three-firm example, demand is sensitive to supply only because of a stock out. For supply sensitivity without stock outs, the game could specify a weighting factor that reserves a fraction of aggregated demand for allocation by order rotation, leaving the balance for GP allocation. The higher the weighting factor, the more sensitive demand would be to supply.
Real-Market Application
The recipes for structural fairness were developed for a game with real markets for resources, products, and banking services. The game was an Internet-based (Pillutla, 2003) multirole, multinational business game whereby customers purchase products sold by virtual firms owned and managed by players. The customers are players purchasing products for their own virtual consumption, and other virtual firms purchasing products as resources for their own production process. In this game, every player is necessarily a consumer, because the game scores players by their consumption. Those who consume more, by purchasing, ergo virtually consuming, products and consuming them more steadily over the duration of the game achieve higher scores. Moreover, every player is free to exercise ownership and executive roles by founding firms, acquiring shares in firms, and accepting executive employment in firms. In the four-quadrant classification system of Crookall, Martin, Saunders, and Coote (1986), the game is computer-assisted, rather than computer-directed, computer-based, or computer-controlled, because the game supports extensive participant-participant interaction and leaves participants in control of outcomes.
The episodes of the game are labeled periods, each period consisting of a batch-processing interval (sub-period 0) when all players’ decisions are collected and processed together, followed by a continuous-processing interval (sub-period 1) when each player’s decision is processed when that decision is submitted (Lainema, 2010). Sub-period 0 of the game is an interval of generally between 5 and 50 seconds when the application suspends players’ actions while it intensively computes production quantities, executes transactions, and updates the status of all parties from the previous period. Sub-period 1, following immediately after, is an interval, varying from 10 minutes to one week depending on administrative setting, during which players can execute actions at will and immediately realize the consequences of their actions.
The execution-to-immediate-realization effect of sub-period 1 is akin to purchasing an airline ticket over the Internet. When the purchaser clicks the execute-purchase button, the full price of the ticket is at once debited to the purchaser’s credit-card account after which the airline’s server responds with a purchase-confirmation message. Ticket bought, money gone. The airline, the purchaser, and the bank thus immediately realize the consequences of the purchaser’s action.
The issue of fairness arises from the game’s double-auction, transaction-based markets (Shubik, 2005; Teach, 1990; Thavikulwat, 1997), which includes a product market, a stock market, and a banking market. In such a market, buyers and sellers submit offers that are matched by a clearinghouse algorithm. A transaction results when an ask-to-sell offer is successfully matched to a bid-to-buy offer. The transaction can arise from a sale either by Firm X to Customer A or by Firm Y to Customer A. Whether Firm X or Firm Y gets the sale matters, because the firm that does not get the sale may have to settle for inferior terms with the next customer in line. Hence, the sequence in which items are transacted is important.
Transactions are executed in the order by which offers are sequenced. A simple way of sequencing offers is by price and time priority. That is, better-price offers are ahead of worse-price offers, and, for same-price offers, earlier-submitted offers are ahead of later-submitted offers, which advantages classes meeting earlier over classes meeting later when more than one class participates in the same administration of the game. To assure fair treatment irrespective of offer-submission time, arrival rotation (Equation 9) was applied to same-price offers in the product and stock markets, replacing time priority.
Arrival rotation is unsuited to the banking market, because of the special character of the banking business. The game’s banking market matches the deposits and loans of players and product firms to the bank offering the highest interest rate on deposits and the lowest interest rate on loans. The issue of fairness arises when two or more banks sets the same highest interest rate on deposits or the same lowest interest rate on loans. This situation differs from the product and stock situations, because the banks are not limited in the quantity of deposits they take or the quantity of loans they extend. Simulating the modus operandi of modern banking, the game arranges for the difference between the deposits and loans of any bank to be lent to or borrowed from other banks, including the central bank, whose capacity to accept deposits and extend loans is unlimited. Accordingly, if customers are assigned to banks based on the bank’s position in the queue, however that position might be determined, the first bank in the queue would get all the business in each period, giving rise to volatile revenues among banks were the first position be rotated among banks from period to period.
To forestall volatility, the queues of both the banks and their customers are fixed and positional rotation (Equation 2) is applied to rotate the customers among the banks. The queue of banks is fixed by basing it on the order in which banks are founded, so banks founded earlier are always ahead in the queue relative to banks founded later. The queue of customers is similarly fixed, by basing it on the order in which players register and product-firms are founded, with players preceding product-firms in the queue. The customers, composed of players and product firms, are matched to banks by applying Equation 2, with j re-defined to be the position of the customer in the customer’s queue. Thus, for the case of N = 3 banks (Bank 0, Bank 1, and Bank 2) in Period i = 1, Equation 2 assigns customer j = 1 to Bank (1 + 1) mod 3 = 2, the third bank. In the next period (i = 2), given the same two queues, the same customer would be assigned to Bank (2 + 1) mod 3 = 0, the first bank, and so forth. This positional rotation procedure distributes the customers evenly among all banks with the highest interest rates on deposits and the lowest interest rates on loans in each period, and rotates the distribution across periods.
The final problem of fairness in the game arises from its multinational nature. The game allows every nation to have its own clearinghouse for products. The problem centers on the question of which nation’s clearinghouse should match offers to trade products, considering that the nations may have trade policies that affect prices, so a higher-price offer in one nation might be a lower-price offer in another nation.
The problem can be resolved in one of two ways. First, the clearinghouse can be fixed in one nation for an entire period, with the resident nation rotated from period to period. Second, the offers of each period can be rotated among the clearinghouses of all nations in every period. The first approach is simpler, but the second one lowers volatility because it spreads international differences across the many transactions of each period. The issue of fairness arises in both approaches.
The game takes the second approach and applies order rotation to the nations, preferred over positional rotation because when N + 1 is a prime number, order rotation assures both complete order fairness and complete rotational fairness, whereas positional rotation assures only complete positional fairness. Even so, in cases when the prime-number condition is not met, order rotation requires skipping, which undermines positional fairness. In these cases, would positional rotation, which assures positional fairness but not order fairness, be better?
The answer is no. To see why, consider the case of three nations (A, B, and C), each with a single resident firm, F1 in A, F2 in B, and F3 in C, as shown in Table 19. For simplicity, let all three nations share the same currency, impose an identical tariff (t > 0) on imports, produce the same non-storable service product, and ask the same price (p). Thus, if t is $1 per unit and p is $2 per unit, then the import tariff of all three nations is the same, at $1 per unit, and the price charged by all three firms also is the same, at $2 per unit. The firms differ in that F1 and F3 each produces one unit of the product each period, whereas F2 produces 3 units of the product each period.
Condition of Firms.
The three firms compete for sales to a single customer who bids each period for 3 units of the product at price ≥ p + t, high enough to assure that the bid will be completely successful irrespective of the customer’s resident nation. Inasmuch as F1 is identical in all respects to F3, a fair rotational method would ensure that both firms realize the same sales.
Table 20 shows the sales resulting from positional rotation over the three-period cycle of three parties, each episode being a period and each party being a nation in this case. The clearinghouse processing sequence of Period 1 is Nation A, followed by Nation B, and followed by Nation C (A-B-C) as shown, because positional rotation of three parties over three episodes requires that for Episode 1 Party A be served first, followed by Party B, and followed by Party C (cf. Table 1).
Sales of Firms With Positional Rotation Across Three Market in One Cycle.
Accordingly, in Period 1, the clearinghouse of Nation A is active first. From the vantage point of Nation A, the asking price of F1 is p, but the asking prices of F2 and F3 are higher because of tariff, effectively p + t for both. So, the clearinghouse matches the customer’s three-unit bid to the ask-offer of F1, which has effectively the lowest asking price. F1 realizes the sale of the one unit it has produced and made available for sale, and the customer’s three-unit bid is reduced to a two-unit bid.
The clearinghouse does what a rational customer would do. In this case, if p = $2 per unit and t = $1 per unit, the rational customer would choose to buy the product from F1, because the purchase would require the expenditure of only $2 per unit. On the other hand, if the customer had chosen to buy the product from either F2 or F3, the customer would have to pay $3 per unit, $2 to the firm selling the product and $1 to the tariff collector of Nation A.
Clearinghouse activity then moves to the second nation of the sequence, Nation B, where the customer’s two-unit bid is matched to the ask-offer of F2, similarly advantaged by the tariff imposed on F3. The three-unit production of F2 is more than enough to meet the customer’s two-unit bid, so F2 realizes sales of two units, leaving one unit unsold. No customer offer remains, so F3 does not sell the one unit that it has produced.
In the second period, clearinghouse activity starts and ends in Nation B, because the three units that F2 has for sale matches exactly the quantity bid. F2 sell three units; the other two firms sell none. In the third period, clearinghouse activity starts in Nation C and ends in Nation B. Each firm sells one unit.
Accordingly, by the end of the three-period cycle, F1 has sold twice as much as F3. Clearly, positional rotation is unfair in this case.
Table 21 shows the sales resulting from order rotation over the four-period cycle of three parties, each party being a nation, plus one dummy party (D) added to the rotation (cf. Table 2) and skipping the dummy nation in the applied sequence. The matching of ask-offers each period to the customer’s bid proceeds as in positional rotation. In Period 1, the sequential order A-B-C is the same as that of the previous case (cf. Table 18), so the result is the same. F1 sells one unit, F2 sells 2 units, and F3 sells none. In Period 2, clearinghouse activity starts in Nation B, where F2 is advantaged over the other firms. F2 sells its entire production quantity of 3 units to the single customer. Demand satisfied the customer buys nothing more, so both F1 and F2 sell nothing. In Period 3, the sequential order C-A-B is again identical to that of the previous case, so the result is the same as that of the previous case. The three firms sell one unit each to the customer. In Period 4, which does not appear in the previous case, clearinghouse activity begins in Nation C, where F3 is advantaged over the other firms. F3 sells its one unit, whereupon clearinghouse activity moves to Nation B, resulting in F2 selling two of the three units that it has produced. Demand satisfied the customer buys nothing more, so F1 sells nothing. The cumulative sales of F1 and F3 over the four periods are now the same, at two units each. Thus, order rotation is fair.
Sales of Firms With Order Rotation Across Three Market in One Cycle.
Conclusion
Positional rotation, order rotation, and arrival rotation are three recipes for fairness in games of more than a single episode. For games wherein the number of parties does not change from episode to episode, judiciously combining positional and order rotation assures both complete positional and complete order fairness within N episodes for N parties when N is even. When N is odd, the same can be similarly assured within 2N episodes. Where the number of parties vary from episode to episode, arrival rotation assures fairness irrespective of the parties’ order of arrival. Proportional- and random-allocation methods are generally inferior methods of assuring fairness, because they require conditions that are more restrictive.
Fairness is an issue of business games with modeled and real markets, less for the former and more for the latter. Broadly conceived, all games with a scoring system that is taken seriously are business games, for business is fundamentally an activity that participants engage in for gain, that is, for a better score. The score might be labeled profit in one game and peace in another, but these are superficial differences. Accordingly, the recipes discussed herein may apply to many games that are nominally not business games.
Further research on fairness in games might proceed towards clarifying the conditions under which each method yields the smallest root-mean-squared difference between the proportion of opportunities required and the proportion of opportunities distributed. For now, the conclusion is that games can be structurally fair, but the structurally fair game must be a multi-episodic game that incorporates fairness into its design.
Footnotes
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