Dilemma 1: Dilemma Games

     For the next four or so weeks, starting today, I shall be blogging, one chapter per weekday, from my book about Dilemma Games.


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        Dilemma

               A New Way To Play

         by Nathaniel Hellerstein

Table of Contents

Introduction

Part 1:  Dilemma Defined

Dilemma Games...................................................................... 4

Dilemma Tournaments............................................................ 6

Dilemma Strategies................................................................. 8

Dilemma Wagers................................................................... 11

Part 2:  Hoyle’s Dilemma

Dilemmizing Games.............................................................. 15

Dilemma Finger-Matching.................................................... 16

Dilemma Tic-Tac-Toe........................................................... 17

Dilemma Dice and Coins....................................................... 19

Dilemma Card Games............................................................ 20

Dilemma Sports..................................................................... 22

Dilemma Video...................................................................... 24

King’s Dilemma..................................................................... 26

Dilemma Checkers................................................................. 29

Dilemma Chess...................................................................... 30

Part 3:  Dilemma Theory

Freeing the Prisoner............................................................... 45

The Shadow of the Future...................................................... 48

Milo’s Trick........................................................................... 50

The Tragedy of the Commons............................................... 53

Predictor’s Paradox................................................................ 56

Mutual Profit.......................................................................... 60

The Unexpected Departure.................................................... 62

                        Chicken.................................................................................. 66

Bibliography....................................................................................... 68

Index.................................................................................................. 71

                                                            Copyright  Nathaniel Hellerstein

Introduction

This book is about games; parlor games, economic games, mathematical games, and political games. These games are “non-zero-sum”; that is, one side’s gain need not be the other side’s loss. Cooperation is possible but vulnerable. This opens up a whole new dimension of play; between competition and cooperation. This book tells how to make peace for fun and profit.

Part 1 of this book defines dilemma games; playing them, scoring tournaments, strategies, negotiation, and wagers.

Part 2 is about dilemmizing standard paper, board, and card games. Games from tic-tac-toe to chess reveal long-hidden truces.

Part 3 covers dilemma theory; how to prosper within a dilemma. (Reciprocity is the key.) It discusses the tragedy of the commons, mutual profit, the unexpected departure, and the game Chicken.

My friends and collaborators in this venture include Lou Kaufmann, Douglas Hofstadter, Tarik Peterson, Stan Tenen, Rudy Rucker, Jim Fournier, and Dick Shoup; without their vital imput, this book would have been impossible.

Love and thanks go to my parents, Earl and Marjorie Hellerstein, of blessed memory, without whom I would have been impossible.

Special thanks go to my dear wife Sherri, without whom I would not have written this.

Finally, due credit (and blame!) goes to myself, for boldly going where game theorists fear to tread.

Part 1:

Dilemma Defined

Dilemma Games

It is proverbial that you cannot compare apples and oranges. The proverb is nonsense, of course; you can compare apples and oranges, but different people will compare them differently.

Consider the case of Alice and Bob. Alice likes oranges more than apples; she’d pay $1 for an apple and $2 for an orange, but all she has is an apple. Bob is just the other way around; he’d pay $1 for an orange and $2 for an apple, but all he has is an orange.

Clearly it would be worth both their while to exchange. Alice and Bob arrange the exchange thus: they put paper bags at opposite ends of a park bench, then walk past each other, pick up the bags, and go.

As you can see, Alice has had some sharp dealings with Bob in the past; so naturally she wonders. What if Bob leaves an empty bag? Then he’d have an apple (worth $2 to him) and an orange (worth $1 to him), for a total of $3 to him; while Alice would get nothing.

Conversely, if Alice tricks Bob by leaving an empty bag, then she’d get $3 worth, and Bob would get $0. If Alice and Bob both leave empty bags, then they’d get $1 each; but if the deal goes through, then both get $2 worth.

0These results are summarized in this table:

                            B

             (A,B)|  nice      mean

                  |        |

              ----|--------|---------|   

             nice |  2,2   |   0,3   |   nice = leave full bag

        A     ----|--------|---------|   

             mean |  3,0   |   1,1   |   mean = leave empty bag

                  |--------|---------|

The table entries describe payoffs for the two players. Note that this is a “non-zero-sum” game; one player’s gain is not necessarily the other player’s loss. Thus we get a dilemma; and indeed this game is (here) called Dilemma.

This is the general Dilemma payoff matrix:

                   B

(A,B)  |  co-op    defect

       |                         W = Win

   ----|--------|---------|      T = Truce

co-op  |  T,T   |   L,W   |      D = Draw

  A         -|--------|---------|      L = Lose;

defect |  W,L   |   D,D   |

      -|--------|---------|       so L < D < T < W;

                                  also W+L = D+T.

If players A and B cooperate, then they both get T, which is more than D, which they would both get if they compete; but if only one cooperates, that one loses big to the other. Both players are tempted to cheat; but if both do, neither wins!

In Dilemma, both parties can benefit if they cooperate, but both are tempted to cheat. This non-zero-sum game poses a dialectical dilemma; a player’s paradox. This little game exemplifies the central dilemma of any society; namely, how to get people to co-operate for mutual benefit, when competitive behavior yields a tactical advantage.