The Curious Case of Zero-Sum, Games, and Monopoly
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"Josh Adelson" <adelson.05@NOSPAM.comcast.net>:
# ...the most time-honored, valuable, and VALID definition
# of a zero-sum game: One you can't win unless every other
# participant loses. And that's that. Next we'll cover
# Nash Equilibrium; the Prisoner's Dilemma....
thanks for the mentions. I'm not sure you're position is
supported, entirely, in the material below. I like
research and don't mind citation/typing. this is not as
clear-cut a case as some of you suggest. enjoy the
discussion below and please offer correction!
funny, even "zero-sum" isn't in my Am Her Dic. or Websters.
Wikipedia (sometimes questionable) indicates that
"zero-sum" was itself developed as part of game theory
and means:
WIK} Zero-sum and non-zero-sum games
WIK}
WIK} In zero-sum games the total benefit to all players in the game,
WIK} for every combination of strategies, always adds to zero (or
WIK} more informally put, a player benefits only at the expense of
WIK} others). Go, chess and poker exemplify zero-sum games, because
WIK} one wins exactly the amount one's opponents lose. Most real-world
WIK} examples in business and politics, as well as the famous
WIK} prisoner's dilemma are non-zero-sum games, because some outcomes
WIK} have net results greater or less than zero. Informally, a gain
WIK} by one player does not necessarily correspond with a loss by
WIK} another. For example, a business contract ideally involves a
WIK} positive-sum outcome, where each side ends up better off than
WIK} if they did not make the deal.
WIK}
WIK} Note that one can more easily analyse a zero-sum game; and it
WIK} turns out that one can transform any game into a zero-sum game
WIK} by adding an additional dummy player (often called "the board"),
WIK} whose losses compensate the players' net winnings.
WIK}
WIK} A game's payoff matrix represents convenient way of
WIK} representation....
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from
http://en.wikipedia.org/wiki/Game_theory#Zero-sum_and_non-zero-sum_games
=========================================================================
thanks for introducing this topic. I don't think I've really
considered it before except in passing, contrasting it with
playing of games as is conventionally meant by the term.
IS "game theory" something that has relevance to *board* games?
possibly not, but game theory enthusiasts like to use 'em.
Wikipedia has:
WIK} Game theory is a branch of applied mathematics that uses models
WIK} to study interactions with formalised incentive structures
WIK} ("games"). It has applications in a variety of fields, including
WIK} economics, international relations, evolutionary biology, political
WIK} science, and military strategy. Game theorists study the predicted
WIK} and actual behaviour of individuals in games, as well as optimal
WIK} strategies. Seemingly different types of interactions can exhibit
WIK} similar incentive structures, thus all exemplifying one particular game.
-----------------------------------------------------------------------------
from
http://en.wikipedia.org/wiki/Game_theory
=========================================
so it sounds like "game" means something else to 'game theorists'
than it does to board gamers, generally. this is confirmed by
academics of game history (rather than game theory): when I
look for 'zero-sum' in Parlett I found no index cite, but
he *does* make a comment on "game theory", and mentions
*Monopoly* too!:
_Theme Games_
Theme games are board games purporting to simulate or
represent some sort of real-life activity. There is an
important difference between simulations and
representations. Simulations, as the term implies, are
designed to enable the participants to exercise as
realistically as possible the the same sort of skills
of judgement and decision-making as are required for the
subject in question. They are not 'games', nor are the
participants 'players', in the usual sense of the words,
since the exercise is designed not for fun, recreation,
or social intercourse, but to provide exercise, practice,
training experimentation, and so on, in an important field
of real-life endeavour. Simulations include war games as
played by the military and business games as played on
weekend management courses, and the sort of subjects
explored in that branch of mathematics called 'game
theory' -- which may relate to the 'real world outside',
but in fact bear very little relation to anything that
goes in [sic] in the (paradoxically) 'real' world of
games inhabited by genuine games enthusiasts.
Genuinely recreational theme games are therefore
better described as representational than as simulatory.
Indeed, the less realistic they are, whether due to the
creative limitations of the inventor or the inherent
limitations of the material, the less representational
they become, and the more symbolic or even purely
nominal. Monopoly, for example, characterizes itself
as 'the property trading game'. No one would regard
it as a simulation, but whether it rates as
representational, symbolic, or purely nominal,
is probably open to conflicting interpretations.
----------------------------------------------------------
"The Oxford History of Board Games", by David Parlett,
Oxford University Press, 1999; p. 348.
==========================================================
which doesn't really nail the lid on the coffin of 'Monopoly
as zero-sum', but it does separate Monopoly somewhat from
the kind of 'game' covered BY 'game theory' of mathematicians.
nonetheless a bunch of these game theorists are using
board games for examples, including Monopoly!
Wiki's entry on 'zero-sum fallacy' is interesting too --
I'll simply note it may be found at
http://en.wikipedia.org/wiki/Zero-sum_fallacy
-- because it seems to be affirming that (ortho-) Chess and
Poker are zero-sum games and yet disputes economics as relevant
for its application. this may confuse the issue, because in the
game of Monopoly not only does the Bank introduce new monies
every time someone passes Go (so potentially an endless supply
may be available in a kind of 'increasing pie-wedge'), but it
also serves to collect the property mortgaged in order to pay
debts at the end of the game when the final players land on
the winner's (likely monopolized, developed) real estate.
my questions in response to the material I collected so
far regarding zero-sum "games" as applied to Monopoly and
those with which it is compared (primarily Chess/Poker) are:
-- would the entire, actual, assets of the losers
need be collected by the winner of Monopoly in
order to effectively qualify it as "zero-sum"?
(i.e. does the conversion of real estate through
the intermediary of the Bank disqualify it?)
-- does the *existence* of the Bank and its potentially-
unending resource disqualify a "zero-sum" assessment?
(i.e. would the winner have to, effectively, own
all of the properties and bankrupt the Bank *too*
in order for Monopoly to be 'zero-sum'?)
-- if the Poker game had a dealer who skimmed some of
the monies from the pot for his (/the House's)
services would this comparably disqualify the
game of Poker as a 'zero-sum game' also??
what if the Poker game had an unknown 'bottom
of the pockets of the players'? wouldn't that
turn Poker into a non-zero-sum game also?
these questions are interesting to me in attempting a clear
understanding of how "game theory" and recreational games
do in fact intersect. many of the WWW sites I can search
on via Google mention ordinary board games as part of their
examples, and even in presenting 'zero-sum' arguments, and
this doesn't help much. I'll try to answer them below.
Principia Cybernetika seems to address the situation
DIRECTLY, so this confusion must come up repeatedly because
of the overlap of terminology between board gamers and
math-heads and the complexity of what is being claimed
and about what:
PC} Zero sum games
PC}
PC} A game is an interaction or exchange between two
PC} (or more) actors, where each actor attempts to optimize
PC} a certain variable by choosing his actions (or "moves")
PC} towards the other actor in such a way that he could
PC} expect a maximum gain, depending on the other's response.
PC} One traditionally distinguishes two types of games.
PC} Zero-sum games are games where the amount of "winnable
PC} goods" (or resources in our terminology) is fixed.
PC} Whatever is gained by one actor, is therefore lost by
PC} the other actor: the sum of gained (positive) and lost
PC} (negative) is zero. This corresponds to a situation of
PC} pure competition.
PC}
PC} Chess, for example, is a zero-sum game: it is impossible
PC} for both players to win (or to lose). Monopoly (if it is
PC} not played with the intention of having just one winner)
WHAT??? does this person mean 'if it IS played with the
intention of having just one winner'?? are you sure this
isn't Principia *Discordia*? ;)
PC} on the other hand, is a non-zero-sum game: all participants
PC} can win property from the "bank". In principle, in monopoly,
PC} two players could reach an agreement and help each other in
PC} gathering a maximum amount from the bank. That is not really
PC} the intention of the game, but I hope I have made the
PC} distinction clear:
I don't think that they have, based on the above! however, this
maximizing of amounts is precisely what I mentioned before, in
that my brother and I used to play to try to 'break the Bank'
and rack up the highest "score" (net wealth) before dealing to
some kind of conclusion whereby one of us broke the other or
the time-limit was reached.
PC} in non-zero-sum games the total amount gained is variable,
PC} and so both players may win (or lose). When they can both
PC} win by cooperating in some way, we might say that their
PC} cooperation creates a synergy.
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from
http://pespmc1.vub.ac.be/ZESUGAM.html
=====================================
that said, many of these websites are supporting the idea
that Monopoly is *not* a "zero-sum game". I'll look once
more to see if anyone describes it more clearly, or with
what appear to be less typos. let me tell you, there is
no ABSOLUTE consensus on this issue, just from a quick look
online -- so misconceptions appear to be somewhat common,
and very likely because of terminological diffraction due
to overlapping terms (with differing meanings.
those who are part of this dispute probably represent a FAQ
issue and should be excused for whatever hostility and
arrogant intractibility which they are bringing to discussion.
therefore the focus on *mistakes* and their correction would
seem valuable in sourcing for this confusing matter. I find
the explanation (which includes many board games) following
to be more convincing than some of those above or online
which contend that Monopoly is zero-sum (there are some!):
GT} Zero Sum vs. Non-Zero Sum - In zero sum games, the total
GT} value of the game stays the same or goes down. In a normal
GT} Poker game, players buy into a game with the same amount
GT} of money to start with.
that answers one of my questions, 'bottomless pockets' would
not be part of 'zero-sum Poker'. all the same, the Monopoly
game does have a set amount of money, but it is not in play
at the same time and some of it and the properties are owned
by the Bank during most if not all of the game (the exception
being the unlikely event that a winner got it all, which is
not a necessary conclusion to Monopoly).
GT} If six players each start out with $50 worth of chips,
GT} then at any point in the progress of the game the total
GT} of player holdings and the pot will equal $300. (In many
GT} poker games, however, people add more money from their
GT} pocket, but we can safely say that that the total amount
GT} of money in the game does not exceed the total worth of
GT} the players in a game).
more fudging!!!
GT} Chess is another zero sum game, because the number of
GT} chess pieces available never goes up.
here I'll inject some (humourous?) objections.
in Chess the *value* of pieces CAN increase. the *number*
of pieces can never go up, but the *value* of them can do
so at the event of the pawn reaching the last rank in its
advance. strictly theoretically one might have *9 Queens
on each side*. this is extremely and probably logically
impossible to achieve because of the way that pawns move
and the fact that they are in each other's way -- even if
both players colluded to make this happen, only 6 of the
pawns on each side could shift files by taking a non-pawn
piece and enable a pawn from each side to proceed unimpeded.
even so, the number of pieces would not increase. why ought
we be paying attention to the number of pieces when in fact
the value of them contributes to the overall win? does the
fact that we're losing site of the WINNING CONDITIONS here
make any difference? no piece might be lost prior to the
checkmate (remember those facile 5-move puzzles??). but a
piece *must* be lost for a pawn to become a queen or any
other greater value by reaching the final rank. does the
fact that the total value *is always owned by someone*
make the difference here, or that number/value ostensibly
(and where it comes to value, falsely) decreases?
GT} Non-zero sum games are those where values can and do
GT} rise. In Monopoly(tm), every time someone passes GO
GT} another $200 in Monopoly money is added to the game.
GT} Othello(tm) (Reversi) and Go are non-zero sum board
GT} games where pieces are added to the board as the
GT} game progresses.
-----------------------------------------------
from
Game Theory
http://members.cox.net/mathmistakes/game_theory.htm
===================================================
this is still minorly confusing, because Chess, at least,
and Checkers, are *not* games in which values cannot rise,
though their piece numbers per se only decrease. Pawns
can in fact become Queens, Rooks, Knights, or Bishops,
and Checkers can become Kings.
perhaps Monopoly is merely mistakenly associated with
zero-sum in the same way that Earth is sometimes
mistakenly associated with anti-entropic forces. the
Monopoly game mistake is made by ignoring the position
and influence of the Bank, which cascades money into
the game at intervals regularly determined by players
passing Go (would removing the Go-$ make it zero-sum?).
the Earth correlate mistake is ignoring the input from
the SUN, whose solar energy continually washes over
the little planet and thereafter eddies and 'pops up'
in life forms and what appear to be anti-entropic
manifestations out of lifeless rock and otherwise
apparently inert liquids. certainly removing solar
energies from the planet would seem to produce a rock
whose only *addition* of any consequence would be the
occasional smash of a meteor through any atmosphere.
thanks again. :)
Rookie-Move
(nagasiva@luckymojo.com)
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