Analysis of the effectiveness of manufacturer-sponsored retailer gift cards in supply chains

Some manufacturers sponsor “free” retailer gift cards to be given to consumers who purchase their products. These gift cards are paid for by the manufacturer and are redeemable on all products at the retailer. We develop a model of such a supply chain. We analyze cases in which the gift cards’ redemption rate is constant or increasing in gift card value. The results indicate that in addition to the redemption rate and consumers’ valuation for gift card dollars, the profitability of manufacturer-sponsored gift cards depends on the average gross margin of the retailer and the type of purchases consumers make with gift cards. Furthermore, we show that under certain conditions, free gift cards will increase the expected profits of the retailer and manufacturer as well as decrease the retail price of the product. These conditions include a retailer with large average gross margin and consumers using gift cards to purchase products they would not buy with cash otherwise. Furthermore, all consumers, including those who do not redeem the gift card, are more likely to benefit from a reduced retail price when their probability of redeeming the gift card after purchase is equal to their estimated redemption probability at purchase time. We show the conditions under which gift cards are more profitable than cash mail-in rebates. We develop an incentive scheme to improve the performance of supply chains with gift cards.     

A Problem in Permutations: the Game of 'Mousetrap'

The game of 'Mousetrap, a problem in permutations, first introduced by Arthur Cayley in 1857 and independently addressed by Cayley and Adolph Steen in 1878, has been largely unexamined since. The game involves permutations of n cards numbered consecutively from 1 to n. The cards are laid out face up in some order and the game is played by counting on the cards, beginning the count with 1. If at any time the number of the count matches the number on the card, this is called a hit and the card is thrown out. The counting begins again with 1 on the next card and returns to the first card when the nth card is reached. Each time a card is hit, that card is removed and the counting starts over at 1. The game continues until all the cards have been hit and thrown out (the player wins) or until the counting reaches n with no cards having been hit (the cards win). The game is re-introduced here and a summary of both Cayley's and Steen's work is presented. Computer programs, written to generate the types of permutations dealt with by Steen, uncovered discrepancies in his work. Further examination of these discrepancies lead to the discovery of a combinatorial pattern of coefficients which Steen was unable to recognize because of his computational errors. Corrected versions of Steen's erroneous formulas are presented.     

Single-suit two-person card play

We consider a two-person constant sum perfect information game, which we call theEnd Play Game, which arises from an abstraction of simple end play positions in card games of the whist family, including bridge. This game was described in 1929 by Emanuel Lasker, the mathematician and world chess champion, who called itwhistette. The game uses a deck of cards that consists of a single totally ordered suit of 2n cards. To begin play the deck is divided into two handsA andB ofn cards each, held by players Left and Right, and one player is designated as having thelead. The player on lead chooses one of his cards, and the other player after seeing this card selects one of his own to play. The player with the higher card wins a “trick” and obtains the lead. The cards in the trick are removed from each hand, and play then continues until all cards are exhausted. Each player strives to maximize his trick total, and thevalue of the game to each player is the number of tricks he takes. Despite its simple appearance, this game is quite complicated, and finding an optimal strategy seems difficult. This paper derives basic properties of the game, gives some criteria under which one hand is guaranteed to be better than another, and determines the optimal strategies and value functions for the game in several special cases.     

A random walk on a circular path

This short note introduces an interesting random walk on a circular path with cards of numbers. By using high school probability theory, it is proved that under some assumptions on the number of cards, the probability that a walker will return to a fixed position will tend to one as the length of the circular path tends to infinity.     

Chaotic maps-based password-authenticated key agreement using smart cards

Password-based authenticated key agreement using smart cards has been widely and intensively researched. Inspired by the semi-group property of Chebyshev maps and key agreement protocols based on chaotic maps, we proposed a novel chaotic maps-based password-authenticated key agreement protocol with smart cards. In our protocol, we avoid modular exponential computing or scalar multiplication on elliptic curve used in traditional authenticated key agreement protocols using smart cards. Our analysis shows that our protocol has comprehensive characteristics and can withstand attacks, including the insider attack, replay attack, and others, satisfying essential security requirements. Performance analysis shows that our protocol can refrain from consuming modular exponential computing and scalar multiplication on an elliptic curve. The computational cost of our protocol compared with related protocols is acceptable.     

Staircase rook polynomials and Cayley’s game of Mousetrap

First introduced by Arthur Cayley in the 1850’s, the game of Mousetrap involves removing cards from a deck according to a certain rule. In this paper we find the rook polynomial for the number of Mousetrap decks in which at least two specified cards are removed. We also find a new expression for the rook polynomial for the number of decks in which exactly one specified card is removed and give expressions for counts of two kinds of Mousetrap decks in terms of other known combinatorial numbers.     

Customer retention: a behavioural model

One of the main problems currently facing CTedit-card issuersis the increasing number of cardholders who are using theircards less often (i.e. attrition) and/or returning their cards(closures). This problem is of particular concern as the totalnumber of credit cards held by consumers is declining (by approximately0.6 per cent per month in 1992) and the number of new applicantsis also running at an all time-low (less than 1 per cent permonth in 1992). Most of the published literature in the broad area of creditcards looks at credit scoring, rather than the need for cardissuers to identify and retain a profitable portfolio of cardcustomers. The overall objective of our research is ‘segmentationfor customer retention’, and this paper aims to identifythe characteristics of card customers who initiate the closureof their accounts Linear discriminant analysis is applied toa sample of approximately 17,000 UK holders of bank credit cards,using various behavioural and sociodemographic variables, andtested on a holdout sample of 10,000 cases.     

Families of pairs of graphs with a large number of common cards

The vertex‐deleted subgraph G?v, obtained from the graph G by deleting the vertex v and all edges incident to v, is called a card of G. The deck of G is the multiset of its unlabelled vertex‐deleted subgraphs. The number of common cards of G and H (or between G and H) is the cardinality of the multiset intersection of the decks of G and H. In this article, we present infinite families of pairs of graphs of order n ≥ 4 that have at least \begin{eqnarray*}2\lfloor\frac{1}{3}(n-1)\rfloor\end{eqnarray*} common cards; we conjecture that these, along with a small number of other families constructed from them, are the only pairs of graphs having this many common cards, for sufficiently large n. This leads us to propose a new stronger version of the Reconstruction Conjecture. In addition, we present an infinite family of pairs of graphs with the same degree sequence that have \begin{eqnarray*}\frac{2}{3}(n+5-2\sqrt{3n+6})\end{eqnarray*} common cards, for appropriate values of n, from which we can construct pairs having slightly fewer common cards for all other values of n≥10. We also present infinite families of pairs of forests and pairs of trees with \begin{eqnarray*}2\lfloor\frac{1}{3}(n-4)\rfloor\end{eqnarray*} and \begin{eqnarray*}2\lfloor\frac{1}{3}(n-5)\rfloor\end{eqnarray*} common cards, respectively. We then present new families that have the maximum number of common cards when one graph is connected and the other disconnected. Finally, we present a family with a large number of common cards, where one graph is a tree and the other unicyclic, and discuss how many cards are required to determine whether a graph is a tree. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 146–163, 2010     

A shuffle that mixes sets of any fixed size much faster than it mixes the whole deck

Consider an n by n array of cards shufled in the following manner. An element x of the array is chosen uniformly at random. Then with probability 1/2 the rectangle of cards above and to the left of x is rotated 180deg;, and with probability 1/2 the rectangle of cards below and to the right of x is rotated 180°. It is shown by an eigenvalue method that the time required to approach- the uniform distribution is between n²/2 and cn² in n for some constant c. On the other hand, for any k it is shown that the time needed to uniformly distribute a set of cards of size k is at most c(k)n, where c(k) is a constant times k³ In(k)². This is established via coupling; no attempt is made to get a good constant. © 1994 John Wiley & Sons, Inc.     

Reconstruction of the Neighborhood Deck of an Ordered Set

We show that the neighborhood deck of an ordered set can be reconstructed from the deck of one point deleted subsets. As a consequence of the above results we reconstruct some maximal cards and present short new proofs of the reconstructibility of ordered sets of width 2 and of the recognizability of N-free ordered sets. We also reconstruct the maximal cards of N-free ordered sets.     

Single-suit two-person card play II. Dominance

This paper studies a two-person constant sum perfect information game, the End Play Game, arising from an abstraction of end play in bridge. This game was described by Emanuel Lasker who called it whistette. The game uses a deck of cards consisting of a single totally ordered suit of 2n cards. The deck is divided into two hands A and B of n cards each, held by players Left and Right, and one player is designated as having the lead. The player on lead chooses one of his cards, and the other player after seeing this card selects one of his own to play. The player with the higher card wins a trick and obtains the lead. The cards in the trick are removed from each hand, and play then continues until all cards are exhausted. Each player strives to maximize his trick total, and the value of the game to each player is the number of tricks he takes. The strategy of this game seems to be quite complicated, despite its simple appearance. This paper studies partial orderings on hands. One partial order recognizes regularities in the value function that persist when extra cards are added to hands. A pair of hands (A ^* , B ^* ) dominates a pair of hands (A, B) for Left, if for any set of extra cards (C ₁, C ₂) added to the deck such that A B (which equals A ^* B ^* ) is a block of consecutive cards in the expanded deck A B {C ₁ , C ₂} the value of (A C ₁, B C ₂) to Left always is at least as much as the value to Left of (A ^* C ₁, B ^* C ₂) both when Left has the lead in both games and when Right has the lead in both games. The main result is that ({4, 1}, {3, 2}) dominates ({3, 2}, {4, 1}). Note that with just four cards the hands {4, 1} and {3, 2} are of identical value — they both take one trick independent of the lead or how the hands are played. The dominance result shows that {4, 1} is preferable to {3, 2} when other cards are present. We show that the dominance relation gives a partial order that is not a total order on hands of 3 or more cards. We also study the total point count ordering, which gives a rough estimate for the value of a hand. We derive upper and lower bounds for the value of a hand with given point count.     

Probabilistic and combinatorial aspects of the card‐cyclic to random insertion shuffle

Consider a permutation σ ∈ S_n as a deck of cards numbered from 1 to n and laid out in a row, where σ_j denotes the number of the card that is in the j‐th position from the left. We study some probabilistic and combinatorial aspects of the shuffle on S_n defined by removing and then randomly reinserting each of the n cards once, with the removal and reinsertion being performed according to the original left to right order of the cards. The novelty here in this nonstandard shuffle is that every card is removed and reinserted exactly once. The bias that remains turns out to be quite strong and possesses some surprising features. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 362–390, 2015     

Avoiding Side-Channel Attacks by Computing Isogenous and Isomorphic Elliptic Curves

Smart cards are being attacked increasingly more, due to their numerous uses and the valuable information stored inside. For this reason, efficient and secure cryptosystems need to be designed. The main problem is that smart cards are resource constrained. Moreover, they are vulnerable to side-channel attacks. In this paper, we use an algorithm to compute side-channel-resistant alternatives to the curves given in the NIST standard and to the new elliptic curves recently presented by Microsoft Research. The algorithm does this by computing isogenous and isomorphic elliptic curves.     

Telecommunications Network Case Study: Selecting a Data Network Architecture

This paper documents a model that was pivotal in deciding which of two architectures should be selected for a frame relay data communications network. The choices are either to continue using the current architecture, or to make a large incremental investment in new equipment which reduces the number of high speed inter-office trunks required to interconnect the switches. The analysis requires optimizing the mix of two types of customer port cards to determine the maximum customer port capacity of a switch. Simple approximations are used to estimate the number of inter-office trunks and trunk cards required. Based in large part on the costs computed by this model, an executive level decision was made to move to the new architecture.     

Weakness and improvement on Wang–Li–Tie’s user-friendly remote authentication scheme

In an open network environment, the remote authentication scheme using smart cards is a very practical solution to validate the legitimacy of a remote user. In 2003, Wu and Chieu presented a user-friendly remote authentication scheme using smart cards. Recently, Wang, Li, and Tie found that Wu–Chieu’s scheme is vulnerable to the forged login attack, and then presented an improvement to eliminate this vulnerability. In our opinion, the smart card plays an important role in those schemes. Therefore, we demonstrate that Wang–Li–Tie’s scheme is not secure under the smart card loss assumption. If an adversary obtains a legal user’s smart card even without the user’s corresponding password, he can easily use it to impersonate the user to pass the server’s authentication. We further propose an improved scheme to overcome this abuse of the smart card.     

On convergent probability of a random walk

This note introduces an interesting random walk on a straight path with cards of random numbers. The method of recurrent relations is used to obtain the convergent probability of the random walk with different initial positions.     

A profit-based scoring system in consumer credit: making acquisition decisions for credit cards

Consumer credit scoring is one of the most successful applications of quantitative analysis in business with nearly every major lender using charge-off models to make decisions. Yet banks do not extend credit to control charge-off, but to secure profit. So, while charge-off models work well in rank-ordering the loan default costs associated with lending and are ubiquitous throughout the industry, the equivalent models on the revenue side are not being used despite the need. This paper outlines a profit-based scoring system for credit cards to be used for acquisition decisions by addressing three issues. First, the paper explains why credit card profit models—as opposed to cost or charge-off models—have been difficult to build and implement. Second, a methodology for modelling revenue on credit cards at application is proposed. Finally, acquisition strategies are explored that use both a spend model and a charge-off model to balance tradeoffs between charge-off, revenue, and volume.     

Extended use of the cards procedure as a simple elicitation technique for MAVT. Application to public procurement in Switzerland

The cards procedure was designed in the early 90 as a simple way to elicit weights for multiple criteria decision analysis outranking methods. It is based on the elicitation of the difference of importance between successive pairs of criteria. We propose to extend its use in two directions:     

Graphs with a common eigenvalue deck

In this paper, we consider graphs whose deck consists of cards (which are the vertex-deleted subgraphs) that share the same eigenvalue, say μ. We show that, the characteristic polynomial can be reconstructed from the deck, providing a new proof of Tutte’s result for this class of graphs. Moreover, for the subclass of non-singular graphs, the graph can be uniquely reconstructed from the eigenvectors of the cards associated with the eigenvalue μ. The remaining graphs in this class are shown to be μ-cores graphs.