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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.

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