Tilburg University. Beat the dealer in Holland Casino's Black Jack van der Genugten, B.B. Publication date: Link to publication

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1 Tilburg University Beat the dealer in Holland Casino's Black Jack van der Genugten, B.B. Publication date: Link to publication Citation for published version (APA): van der Genugten, B. B. (). Beat the dealer in Holland Casino's Black Jack. (Research memorandum / Tilburg University, Department of Economics; Vol. FEW ). Unknown Publisher. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research You may not further distribute the material or use it for any profitmaking activity or commercial gain You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date:. nov.

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3 BEAT THE DEALER IN HOLLAND CASINO'S BLACK JACK ~ ~I B.B. van der Genugten FEW ~ijga.i~ ~ ~ií~c. t (JC~ ~~Gft ~.~ Ne fl~u ~. F~a.c.~j Communicated by Dr. P.E.M. Borm

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5 BEAT THE DEALER IN HOLLAND CASINO'S BLACK JACK B.B. VAN DER GENUGTEN' Abstract This article concerns the game of Black Jack played according to the rules of the foundation rholland Casino's in the Netherlands. It is shown that the strategies based on Thorp's original tencount system lead to a loss by the players. However, it is also shown that these strategies can be adapted in such a way that players gain and thereby can beat the dealer. The methods used for deriving these strategies are strongly based on extensive simulation and combinatorial probability calculus. Keywords Black Jack, Holland Casino's, simulation, combinatorial probability. AMS classi8cation: E, 8U, C. 'Department of Econometrics, Tilburg University, Tilburg, The Netherlands

6 Introduction The golden days for Black Jack players were in the sixties after the publication of Thorp's book () "Beat the Dealer". Based on the Las Vegas rules in those days, this book contained relatively simple strategies which gave players an advantage over the house. Thereafter, casinos took counter measures and changed the rules. These rules vary strongly over casinos but all have one goal in common: to give back the advantage to the house. However, in the second edition of his bestseller, Thorp states that in many cases players can still obtain a slight advantage if they follow his strategies. And that's what professional players still seem to do today. The rules in the casinos run by the foundation "Holland Casino's" in the Netherlands are exactly the same in all cities (Amsterdam, Rotterdam, Groningen, Scheveningen, Zandvoort, Breda, Nijmegen, Valkenburg and Eindhoven). They are typically Dutch in so far that the combination of variations does not appear elsewhere. This raises the first important question: what would happen if we used Thorp's strategies when playing BJHC (Black Jack in Holland Casino's)? The answer is simple: a disaster. We would loose every finite amount of money with probability one. So a second question arises: can we play BJHC in practice such that we really have an advantage and so are able to beat the dealer? The answers is in principle aff'irmative. These questions and their answers are the theme of this article. Rules of BJHC and strategies of players The general rules of Black Jack (also known as: Twenty One) are extensively described in Thorp () or Epstein (). Here we emphasize the differences with BJHC. BJHC is a card game that is played with a maximum of players. Cards are dealt by a dealer, a member of the house. A complete stock of cards consists of decks of cards each ( cards). It is the starting stock for the first game. The remainder of the cards after the first game becomes the playing stock of the second game and so on. If the remaining stock after a game falls below the size of decks ( cards), then all cards are reshuflled and the next game starts with the starting stock. We call a sequence between two successive reshu(ilings a rowgame. All rowgames are mutually independent. Players gain or lose by betting. The minimum bet is (for a cheap table j,) and

7 the maximum bet is usually (corresponding to f,). Cards are always dealt face up. So, at least in theory, every player can know the composition of the stock at any stage of the game. Face cards have the value ; nonface cards have their indicated value. An ace can be counted as or. A hand of cards is called soft if it contains an ace that can be counted as without exceeding the total. Other hands are called hard. We call the maximum total of a hand the sum. It is the goal of a player to get a sum as close as possible to but never exceeding by drawing (asking the dealer for cards one after another) or standing (requesting no more cards) at the right moment. He busts (loses) if his (hard) sum exceeds. Standing on (soft or hard) is obligatory. If in a game at least one player stands, then the dealer has to draw cards, too. He has no choice: he draws on sums C, stands on sums ~ (hard or soft) and busts on a sum ~. If a player and the dealer both stand, then the game is lost for the one holding the smallest sum. The combination Black Jack (A,) beats any other sum of. Equal sums give a draw. A winning player gains an amount x his original bet and even ~ x if he wins with Black Jack. A losing player loses his original bet. In case of a draw a player neither gains nor loses; his bet is returned. Depending on the state of the game, players have the following extra options: insurance, even money, doubledown and split. Furthermore an extra bonus equal to the original bet is gained with a hand of three sevens. We describe in detail a game together with the decisions points of the players. The game starts with the betting of players (decision F: Bet). After that the dealer deals one card to each of the players and to himself (the dealer card). Then a second card is dealt to each of the players (not yet to the dealer). So every player starts with one pair of cards. If the dealer card is an ace, players may ask for insurance against a dealer's Black Jack (decision E: Insurance). A player insures with an amount x his original bet. If the dealer then gets Black Jack, the player gains x his insurance. He otherwise loses this insurance. Next, players, one after another, request cards from the dealer. If the dealer card is an ace and the player has Black Jack, he may ask for even money (decision D: EvenMoney). In that case he immediately gets his gain which will be reduced to x his bet.

8 If both cards of the pair have the same value, a player may split those cards (decision C: Split). Then the dealer adds to both cards one new card and the player continues with both pairs separately. To the additional pair a new bet equal to the original bet must be added. Repeated splitting is permitted without any restrictions. However, with pairs obtained by splitting aces no íurther normal drawing is allowed. For split pairs Black Jack does not count and three sevens no longer give an extra bonus. If a pair has a hard sum of,, or a soft sum of,, doubling down is permitted (decision B: DoubleDown). Then the player doubles his original bet, draws one card and stands. Finally, as long as a player did not stand or bust he can choose between drawing or standing (decision A: Stand~Draw). A particular strategy of a player must specify at each appropriate stage of the game the decisions AF. For the decisions AC, the player's sum, the dealer card and the current stock should be taken into account. For the decisions DF, the information of the current stock is sufficient. Since the number of possible stocks is very large, a strategy cannot be completely tabulated. A very simple strategy for players is to mimic the dealer. Since players have to draw cards first they are at a disadvantage. Table below shows that such players should change to pure gambling games like French or American roulette. However, with the so called basic strategy described in section we see that it is possible to reduce the loss. Even with such a simple strategy Black Jack becomes at once the most attractive casino game to play.

9 Table. Gain of some Cgames Twin Roulette ~. qo Black Jack mimic dealer. Plo French Roulette composite simple ~. qo 8~88.q ~.oi'o American Roulette Black Jack basic strategy.~ Construction of the optimal strategy We consider here the determination of the optimal decisions for a given playstock. The basic information is the standdistribution of the dealer given his dealer card. Table below provides this distribution for the starting stock minus this card. Table. Standdistribution dealer in q(stock: starting stock dealer card) 8 BJ bust A From this table we see that the probability of the dealer standing at starting with a dealer card equals ~. With table the expected gain of a player of standing at a particular sum can be calculated straightforwardly. This gain should be compared with the gains of other de

10 cisions. A) Draw over Stand. As an approximation for the ca.lculations we assume that cards are drawn with replacement. Then the gain of drawing instead of standing can be determined backwards. We start with standing on H (hard sum ). From this we can calculate the gain of drawing on H and therefore also the optimal choice and the corresponding optimal gain. The order is H, HO,..., H, H and S(oft), H and S,..., H and. The case of H (the so called situation), in which the bonus for drawing three sevens can be obtained by drawing on (,), should be distinguished from the other H possibilities. B) DDown over Stand~Draw. By conditioning to the card to be drawn (with replacement) after double down the gain of double down can be calculated from the table of gains for standing. By comparing this gain with the results obtained under A the optimal decision and the corresponding gain can be calculated. C) Split over Stand~Draw~DDown. By conditioning to the card that gives the new pair after splitting, the gain of splitting can be calculated from the table obtained under B. For H, the special situation should be taken into account. A special problem forms repeated splitting. Just drawing with replacement overestimates the effect too much. This has been solved in the following way: stop the splitting of a pair if exactly the same card is drawn again. Note that the special rule for splitting aces complicates the calculations further. By comparing the gain with the table of results obtained under B the optimal decision and the corresponding gain can be calculated again. The following table gives some examples for the startstock without the dealer card. The gains refer to a minimum bet.

11 Table. Optimal choice (stock: starting stock dealer card; bet ) (x impossible) Sum Dealer card Stand Draw DDown SPlit Opt H.. x x S S. f. x x D H.8. x f. SP H 8.. x.8 D H. f. }. x DD H. F.8 f.. DD With a player's H and a dealer card the gain for standing is. and for drawing.. So the optimal decision is standing with optimal gain.. For a player's S the optimal decision is drawing with optimal gain f.. With a player's H and a dealer card, the optimal choice between standing and drawing is drawing with optimal gain.. However, for the special card combination (,) splitting is possible too. In that case the optimal decision is splitting with gain f.. For H doubling down is possible with two cards. With a dealer card the optimal decision in that case is doubling down with optimal gain ~.. For H doubling down as well as splitting is possible with two cards. With a dealer card the optimal decision in that case is doubling down with optimal gain }.. D and E) Even money and Insurance. According to Thorp (), we define for a given playing stock the Tratio as follows:, number of nontens number of tens For the starting stock T()~.. It is easy to see that we should choose for even money and also for insurance if and only if TG.

12 8 F) Bet size. From the calculations of the tables under AE and by conditioning to the player's cards it is possible to calculate the expected player's gain for a playstock. For the starting stock we get the gain. with a bet. It will be clear that for a given playing stock the optimal bet is the maximum if the gain is positive and the minimum if this gain is negative. Due to the very large number of possible playstocks it is not possible to produce a list of tables with optimal decisions and gains for all stocks. It is only possible to construct an algorithm that gives such a table for a particular stock. By means of a simulation program that generates the game results for the players for any choice of their strategies the overall gain for every set of specific strategies can be determined. In particular this is possible for tl~e case that all players follow the optimal strategies as described above. We found with players an overall mean gain per game of f. (half length of ~ confidence interval.) based on a simulation run of rowgames (about games). The algorithm took too much computer time to get this gain more accurate. However, the result is promising enough to look further for good strategies that should give a positive gain and which can be used in practice. It is also clear that the golden days of Black Jack are over for players in Holland Casino's because the observed gain of f. is really quite small with respect to the maximum bet. The basic strategy The construction of this strategy is based on the following idea: calculate with the algorithm described above a table with the optimal decisions for the starting stock and use this table for every playing stock. This gives the table below. There is no difference between a situation and other H sums.

13 Table. Basic Strategy BJIIC Pair Splitting ( split Dealercard A 8 Pair A A Doubling ( doubledown Dealercard A Sum H H H S S Minimal Standin~ Sums A 8 Hard Soft Dealercard Sum Extension: Poor betting Class I Bet l

14 So for the decisions stand, draw, double down and split according to A,B,C we use always this table. With a dealer card we split with (,), double down with a hard sum, draw with a soft sum but stand with a soft sum 8. For the starting stock the Tratio is.. So we never take even money and insurance. The extension for poor betting is discussed in the next section. Using the simulation program it turns out that the overall gain per game of a player with a minimum bet is. (see also table ). Therefore this is also the optimal bet. With this basic strategy we are not able to beat the dealer: we are beaten by him. The extended basic strategy (I) Without counting cards it seems that there is no better strategy than the basic strategy. However, if one is prepared to count then there are possibilities to raise the gain to a positive value. In this section we describe an extension of the basic strategy which leads to a positive gain. In the next two sections we describe strategies with even a higher positive gain. It should be kept in mind that Black Jack is played very fast. Therefore counting strategies must be simple, otherwise they cannot be applied in practice. Perhaps the most simple counting system is Thorp's tencount system. For this we simply count the tens and the nontens. So at any moment we can calculate the Tratio. Decisions are based on that ratio. It appears that stocks which are rich of tens (i.e. have a low Tratio) are favourable for the players. The class division in table below, which was constructed by trial and error, gives a rough indication. Stocks in class (T.) are híghly unfavourable for player. Stocks in class (T between. and.) are slightly unfavourable. Stocks in class with TG. are highly favourable. The frequencies of the stocks have been determined by simulation. Note that 8 oi'o of the stocks fall into the unfavourable classes, and, and only PI'o in the favourable ones.

15 Table. Tratio classes Class Tratio Gain Freq (~o) ~..... ~.. f.8. ff..8 f~ C. f f~ For decisions AC (stand, draw, double down and split) the extended basic strategy is the same as the basic strategy. For decisions DE (even money and insurance) we can make the theoretically optimal decision since we know the Tratio (take even money and insurance iff TC). For decision F(betting) we take the minimum bet if T falls in a class GO and the maximum bet if T belongs to a class ~. With this extended basic strategy the overall gain per game is positive: f.. So this strategy beats the dealer provided that the player's capital is large enough. Therefore we call this betting strategy the strategy for the rich player. For players with a moderate amount of money the drawback is that they have a high probability of getting ruined. Therefore for poor players we consider another criterion different from maximizing the expected gain. We want to choose the bets in the Tclasses in such a way that the probability of ever getting ruined is minimized (given a certain finite starting capital). Given the frequencies f;, the expected gains g; and the bets b; for all classes i we can estimate this ruin probability using the Waldapproximation for the probability of absorbation in a random walk with positive drift and onesided absorbing barrier (see e.g. Ferguson ()). For the distribution of a single step we take a discrete distribution in points b; and fb; with corresponding probabilities Z f; ( g;) and Z f; ( b g;) for all i. By solving an equation in terms of the moment generating function (e.g. with a numerical NewtonRaphson iteration procedure), this probability can be calculated. By minimizing this ruin probability with respect to the ; we find the optimal bets.

16 The last part of table contains the optimal bets. It appears that the optimal solution hardly depends on the starting capital. However, the ruin probability itself does. Table. Structure of gain with the basic strategy (incl. insurance and even money) Class freq(qo) Gain Bet Rich Poor Thorp..... f.. ~.. f.8. ~.. f. f. f.. Total gain. The overview of table for the extended basic strategy contains the optimal bets for the poor player. He should bet if the Tratio falls in class. The overall gain using this betting strategy is f.. Of course this gain is lower than that for the rich player. Table contains also approximately the bets suggested in Thorp (). We see that the bets in the high classes are too low for obtaining a positive gain. Therefore application of this strategy leads to a ruin probability of for all finite starting capitals. The professional strategy (III~ The professional strategy described in this section can be compared with Thorp's complete tencount system adapted for the Holland Casino's rules. It is rather difficult to apply this strategy in practice since one has to learn long tables by heart. The construction of the tables is very cumbersome. We take the starting stock and add or delete tens, thereby traversing the range of the Tratios with very special stocks. We calculate the optimal decisions AC for each Tratio and make one concise table with Tratio bounds.

17 Table. Professional TenCount Strategy BJIíC (III) Pair Splitting ( Split if T G, if underlined: T ) Dealercard A 8 Pair A A ~ ~. ~ ~, ~ Doubling (DDown if T G) Dealercard A 8 H H H S Sum

18 Standing (Draw if T ~) Dealercard A 8 H H H H H H. H8 S S S..~... Sum Poor betting Class Bet Table gives the result. So with T. and a dealer card we split with (,), we do not split with (,), we double down with lí and do not double down with H, we draw with H and stand with H. With T.8 (class ) we bet as a rich player and we bet as a poor player.? The advanced strategy (II) The professional strategy is difí~icult to apply in practice. Therefore we constructed an approximation, the advanced strategy, which is almost equally efficient and much easier to use. We make for each class a separate table of decisions and combine those tables to obtain one concise table. This table is only a little bit more complicated than that of the basic strategy.

19 We constructed the tables for the advanced strategy by simulation with the professional strategy. We counted for each table entry the number of the positive decisions and the negative decisions obtained with the professional strategy. For the table entry of the advanced strategy we took the one which was the largest.

20 Table 8. Advanced Strategy (II) (see table for class numbers) Pair Splitting ( split) Dealercard A 8 Pair A A c ~ ~ ~ ~ '~ 8 8 i): for not split if T belongs to a class ~.

21 Doubling ( doubledown) Dealercard A H ) ~ H H ~ ~ ~ ~ 8 Sum Dealercard A Minimal Standine Sums 8 (~ ) ( ) (C ) (,) Sum Hard (~ ) ( ) (~ ) (~ ) (~ ) (, ) (c ) (C) (G) (c) (C) Soft 8 (~ ) 8 (G ) (CO) Poor betting Class Bet 8 8

22 8 Table 8 gives the result. With (,) and a dealer card we split if T falls into a class ~ and do not split if T belongs to a class G. Wíth (,) and a dealer card 8 we do not split unless the possibility of obtaining three sevens is lost and T belongs to a class. 8 COMPARISON AND CONCLUSIONS In table the extended basic strategy I, the advanced strategy II and the professional strategies III are compared. The poor betting system varies with these strategies and the corresponding gains are indicated under `Bet'. The rich and Thorp's betting system do not vary with the strategies; they are specified in table. Table shows that the advanced strategy II with the poor betting system leads to a gain of ~.8 per game. The corresponding mean bet per game is.8. This results in a gain percentage per bet of ~.qo. With a starting capital of units the probability of ever being ruined with this playing method is qo.

23 Table. Comparison strategies (IIII). I II III Class freq(oi'o) Bet Bet Bet Rich Gain f. ~.8 f. Bet... Plo f. f. f. Cain ~O.OOG f.8 f.8 Bet P!o f. f. b. Gain... Bet... Plo... Ruin Ruin Ruin Poor Thorp Capital Observe that the difference in gain for rich players between II and III is very small but that both strategies give a substantial improvement of the extended basic strategy. The betting strategies of the poor players lcad almost to the same gain. However, now the difference appears in the ruin probabilities. The largest differences appear with starting capitals around. Note that also for strategies II and III Thorp's betting system leads to a loss.

24 If the reader has ambitions to become a professional player he should not have high expectations about his salary. With working hours a year and fast playing of games per hour at a cheap table with minimum bet f, the expected gain for a poor player is.8 x x x f, f. This salary for rich players becomes f 8 ( for I), f ( for II) and f ( for III). So for a reasonable annual salary he should switch to an expensive table with minimum bet of f. However, he should then look for a millionaire as a companion in order to avoid the risk of being ruined. Acknowledgement. The author wishes to thank Peter Borm for his valuable comment.

25 References Thorp, E. () Beat The Dealer, Revised Edition R.andom House, New York. Epstein, R.A. () The Theory of Gambling and Statistical Logic Academic Press, New York. Ferguson, T.S. () Mathematical Statistics: a decision theoretic approach Academic Press, New York.

26 IN REEDS VERSCHENEN F.G. van den Heuvel en M.R.M. Turlings Privatisering van arbeidsongeschiktheidsregelingen Refereed by Prof.Dr. H. Verbon J.C. Engwerda, L.G. van Willigenburg LQcontrol of sampled continuoustime systems Refereed by Prof.dr. J.M. Schumacher J.C. Engwerda, A.C.M. Ran á A.L. Rijkeboer Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation A~lA Q. Refereed by Prof.dr. J.M. Schumacher Jacob C. Engwerda The indefinite LQproblem: the finite planning horizon case Refereed by Prof.dr. J.M. Schumacher GertJan Otten, Peter Borm, Ton Storcken, Stef Tijs Effectivity functions and associated claim game correspondences Refereed by Prof.dr. P.H.M. Ruys Jack P.C. Kleijnen, Gustav A. Alink Validation of simulation models: minehunting casestudy Refereed by Prof.dr.ir. C.A.T. Takkenberg 8 V. Feltkamp and A. van den Nouweland Controlled Communication Networks Refereed by Prof.dr. S.H. Tijs A. van schaik Productivity, Labour Force Participation and the Solow Growth Model Refereed by Prof.dr. Th.C.M.J. van de Klundert J.J.G. Lemmen and S.C.W. Eijffinger The Degree of Financial Integration in the European Community Refereed by Prof.dr. A.B.T.M. van Schaik J. Bell, P.K. Jagersma Internationale Joint Ventures Refereed by Prof.dr. H.G. Barkema Jack P.C. Kleijnen Verification and validation of simulation models Refereed by Prof.dr.ir. C.A.T. Takkenberg Gert Nieuwenhuis Uniform Approximations of the Stationary and Palm Distributions of Marked Point Processes Refereed by Prof.dr. B.B. van der Genugten

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28 lli Ton Geerts Regularity and singularity in linearquadratic control subject to implicit continuoustime systems Communicated by Prof.dr. J. Schumacher Ton Geerts Invariant subspaces and invertibility properties for singular systems: the general case Communicated by Prof.dr. J. Schumacher 8 Ton Geerts Solvability conditions, consistency and weak consistency for linear differentialalgebraic equations and timeinvariant singular systems: the general case Communicated by Prof.dr. J. Schumacher C. Fricker and M.R. Jaïbi Monotonicity and stability of periodic polling models Communicated by Prof.dr.ir. O.J. Boxma Ton Geerts Free endpoint linearquadratic control subject to implicit continuoustime systems: necessary and sufficient conditions for solvabili ty Communicated by Prof.dr. J. Schumacher Paul G.H. Mulder and Anton L. Hempenius Expected Utility of Life Time in the Presence of a Chronic Noncommunicable Disease State Communicated by Prof.dr. B.B. van der Genugten Jan van der Leeuw The covariance matrix of ARMAerrors in closed form Communicated by Dr. H.H. Tigelaar J.P.C. Blanc and R.D. van der Mei Optimization of polling systems with Bernoulli schedules Communicated by Prof.dr.ir. O.J. Boxma B.B. van der Genugten Density of the least squares estimator in the multivariate linear model with arbitrarily normal variables Communicated by Prof.dr. M.H.C. Paardekooper René van den Brink, Robert P. Gilles Measuring Domination in Directed Graphs Communicated by Prof.dr. P.H.M. Ruys Harry G. Barkema The significance of work incentives from bonuses: some new evidence Communicated by Dr. Th.E. Nijman

29 V Rob de Groof and Martin van 'fuijl Commercial integration and fiscal policy in interdependent, financially integrated twosector economies with real and nominal wage rigidity. Communicated by Prof.dr. A.L. Bovenberg 8 F.A. van der Duyn Schouten, M.J.G. van Eijs, R.M.J. Heuts The value of information in a fixed order quantity inventory system Communicated by Prof.dr. A.J.J. Talman E.N. Kertzman Begrotingsnormering en EMU Communicated by Prof.dr. J.W. van der Dussen A, van den Elzen, D. Talman Finding a Nashequilibrium in noncooperative Nperson games by solving a sequence of linear stationary point problems Communicated by Prof.dr. S.H. Tijs Jack P.C. Kleijnen Verification and validation of models Communicated by Prof.dr. F.A. van der Duyn Schouten Jack P.C. Kleijnen and Willem van Groenendaal Twostage versus sequential samplesize determination in regression analysis of simulation experiments Pieter K. Jagersma Het management van multinationale ondernemingen: de concernstructuur A.L. Hempenius Explaining Changes in External Funds. Part One: Theory Communicated by Prof.Dr.Ir. A. Kapteyn J.P.C. Blanc, R.D. van der Mei Optimization of Polling Systems by Means of Gradient Methods and the PowerSeries Algorithm Communicated by Prof.dr.ir. O.J. Boxma Herbert Hamers A silent duel over a cake Communicated by Prof.dr. S.H. Tijs Gerard van der Laan, Dolf Talman, Hans Kremers On the existence and computation of an equilibrium in an economy with constant returns to scale production Communicated by Prof.dr. P.H.M. Ruys 8 R.Th.A. Wagemakers, J.J.A. Moors, M.J.B.T. Janssens Characterizing distributions by quantile measures Communicated by Dr. R.M.J. Heuts

30 V J. Ashayeri, W.H.L. van Esch, R.M.J. Heuts Amendment of HeutsSelen's Lotsizing and Sequencing Heuristic for Single Stage Process Manufacturing Systems Communicated by Prof.dr. F.A. van der Duyn Schouten 8 H.G. Barkema The Impact of Top Management Compensation Structure on Strategy Communicated by Prof.dr. S.W. Douma 8 Jos Benders en Freek Aertsen Aan de lijn of aan het lijntje: wordt slank produceren de mode? Communicated by Prof.dr. S.W. Douma 8 Willem Haemers Distance Regularity and the Spectrum of Graphs Communicated by Prof.dr. M.H.C. Paardekooper 8 Jalal Ashayeri, Behnam Pourbabai, Luk van Wassenhove Strategic Marketing, Production, and Distribution Planning of an Integrated Manufacturing System Communicated by Prof.dr. F.A. van der Duyn Schouten 8 J. Ashayeri, F.H.P. Driessen Integration of Demand Management and Production Planning in a Batch Process Manufacturing System: Case Study Communicated by Prof.dr. F.A. van der Duyn Schouten 8 J. Ashayeri, A.G.M. van Eijs, P. Nederstigt Blending Modelling in a Process Manufacturing System Communicated by Prof.dr. F.A. van der Duyn Schouten 8 J. Ashayeri, A.J. Westerhof, P.H.E.L. van Alst Application of Mixed Integer Programming to A Large Scale Logistics Problem Communicated by Prof.dr. F.A. van der Duyn Schouten 8 P. JeanJacques Herings On the Structure of Constrained Equilibria Communicated by Prof.dr. A.J.J. Talman

31 V IN REEDS VERSCHENEN 88 Rob de Groof and Martin van Tuijl The TwinDebt Problem in an Interdependent World Communicated by Prof.dr. Th. van de Klundert 8 Harry H. Tigelaar A useful fourth moment matrix of a random vector Communicated by Prof.dr. B.B. van der Genugten Niels G. Noorderhaven Trust and transactions; transaction cost analysis with a differential behavioral assumption Communicated by Prof.dr. S.W. Douma Henk Roest and Kitty Koelemeijer Framing perceived service quality and related constructs A multilevel approach Communicated by Prof.dr. Th.M.M. Verhallen z Jacob C. Engwerda The Square Indefinite LQProblem: Existence of a Unique Solution Communicated by Prof.dr. J. Schumacher Jacob C. Engwerda Output Deadbeat Control of DiscreteTime Multivariable Systems Communicated by Prof.dr. J. Schumacher Chris Veld and Adri Verboven An Empirical Analysis of Warrant Prices versus Long Term Call Option Prices Communicated by Prof.dr. P.W. Moerland A.A. Jeunink en M.R. Kabir De relatie tussen aandeelhoudersstructuur en beschermingsconstructies Communicated by Prof.dr. P.W. Moerland M.J. Coster and W.H. Haemers Quasisymmetric designs related to the triangular graph Communicated by Prof.dr. M.H.C. Paardekooper Noud Gruijters De liberalisering van het internationale kapitaalverkeer in historischinstitutioneel perspectief Communicated by Dr. H.G. van Gemert 8 John Gártzen en Remco Zwetheul Weekendeffect en dagvandeweekeffect op de Amsterdamse effectenbeurs? Communicated by Prof.dr. P.W. Moerland Philip Hans Franses and H. Peter Boswijk Temporal aggregration in a periodically integrated autoregressive process Communicated by Prof.dr. Th.E. Nijman

32 V René Peeters On the pranks of Latin Square Graphs Communicated by Prof.dr. M.H.C. Paardekooper Peter E.M. Borm, Ricardo Cao, Ignacio GarcíaJurado Maximum Likelihood Equilibria of Random Games Communicated by Prof.dr. B.B. van der Genugten Prof.dr. Robert Bannink Size and timing of profits for insurance companies. Cost assignment for products with multiple deliveries. Communicated by Prof.dr. W. van Hulst M.J. Coster An Algorithm on Addition Chains with Restricted Memory Communicated by Prof.dr. M.H.C. Paardekooper Ton Geerts Coordinatefree interpretations of the optimal costs for LQproblems subject to implicit systems Communicated by Prof.dr. J.M. Schumacher

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