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For example, in the upper-left rectangle, where both firms charge 3 per bag, they each earn a daily profit of 1,000 In contrast, if they both charge 2 per bag, their individual daily profits fall to only 800, as you can see in the bottom-right rectangle Obviously, if these two firms collude and charge 3 rather than 2 per bag, they can raise profits by 200 per day each The joint monopoly profit that they can earn totals 400 per day ( 200 each) Their problem is to form a cartel that does, in fact, get them this monopoly profit Such a cartel is hard to maintain, however, because the temptation to cheat is ever-present For example, the other two rectangles (the upper-right and bottom-left) show what happens if one firm charges 3 while the other charges 2.

Proof The rst two parts of the theorem follow immediately from the de nitions of the functions itemsT and tidsT To prove part (iii) let k E be a transaction identi er Then, the item set T (e) includes itemsT (E), by the de nition of itemsT (E) By part (ii), tidsT (T (e)) tidsT (itemsT (E)) Since e tidsT (T (e)) it follows that e tidsT (itemsT (E)), so E tidsT (itemsT (E)) The argument for part (iv) is similar Corollary 1 Let T : D P(I) be a transaction data set and let I : P(I) P(I) and D : P(D) P(D) be de ned by I(H) = itemsT (tidsT (H)) for H P(I) and D(E) = tidsT (itemsT (E)) for E P(D) Then, I and D are closure operators on I and D, respectively Proof Let H, H be two subsets of I such that H H .

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By part (ii) of Theorem 4 we have tidsT (H ) tidsT (H); part (i) of the same theorem yields I(H) = itemsT (tidsT (H)) itemsT (tidsT (H )) = I(H ), so I is monotonic The proof of monotonicity for D is similar Since E tidsT (itemsT (E)), by part (i) of Theorem 4 we have itemsT (tidsT (itemsT (E))) itemsT (E) On the contrary, by the expansiveness of I we can write itemsT (E) itemsT (tidsT (itemsT (E))), which implies the equality itemsT (tidsT (itemsT (E))) = itemsT (E) for every E P(D) This, in turn means that tidsT (itemsT (tidsT (itemsT (E)))) = tidsT (itemsT (E)), which proves that D is idempotent The proof for the idempotency of I makes use of the equality tidsT (itemsT (tidsT (H))) = tidsT (H) and is similar; we omit it.

Hopefully, such numbers provide us a rudimentary basis for beginning to comprehend just how big occupational fraud is today The bad news is that we may not even have all of it counted, even at the $600 billion level The 2001 National Retail Security Survey conducted by the University of Florida reported that US retailers lost more than $323 billion in inventory shrinkage25 Some of that loss was caused by human error, damage, and theft by outsiders But if only 10 percent was caused by theft by employees, that amounts to $32 billion dollars larger than all of major league baseball Informed observers from the NWCCC estimate the real percentage of such losses caused by the fraudulent actions of insiders to be about 40 to 45 percent about five times the size of all of major league baseball!26 Even figures this striking may be substantially incomplete.

Closed sets of items, that is, sets of items H such that H = I(H), can be characterized as follows: Theorem 5 Let T : {1, , n} P(I) be a transaction data set A set of items H is closed if and only if for every set L P(I) such that H L, we have suppT (H) > suppT (L) (72) (71).

The firm charging 2 steals business away from the 3 firm and earns a much higher profit Indeed, the firm charging 2 per day earns a profit of 2,000 per day while the firm charging 3 per day earns a profit of only 500, because this firm is losing business to its more cheaply priced competitor If you look at each firm s incentives, you notice that the dominant strategy for each firm is to charge the lower price, 2 per bag To see this, look at Jazzy Drops s pay-offs If Fruity Rock charges 3 per bag, the better thing for Jazzy Drops to do is to charge 2 per bag You can see this by comparing Jazzy Drops s pay-off of 1,000 per day in profits in the upper-left rectangle with its pay-off of 2,000 per day in the bottom-left rectangle.

Here are a few more examples:

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