Phase I: Monopoly
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Parker Brothers introduced Monopoly in 1936, and the game is still
wildly popular today. Its longevity suggests a well-balanced
design, and the game is thus a good choice for analysis. Though the
game is
highly dependent on luck, every player has pet strategies. Whether
any strategies statistically increase a player's success is the subject of
the first phase of this project. Additionally, we will evaluate
the effect of changing typical game parameters, such as the amount of cash
players get at the beginning of the game, and how much they get for passing
Go.
Game Engine
Using a Monopoly game engine written by student Cem Cebenoyan
that simulates computer-driven players and prints output to a log file,
we are analyzing the effects of different player "personalities" on game
outcome. Input to the engine includes: number of iterations,
number of players, typical game data such as property rents, and a list
of AI strategies for each player.
Unlike the standard board game, the computer model has
the following modifications:
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An artificial ceiling of $100,000. When any player
hits this limit, the game is declared winnerless and terminated.
The ceiling is necessary to prevent the infinite games created by particular
strategies.
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No building shortage. Players still build up to one
hotel per property, but the bank never runs out of materials.
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Simplified auctions. To avoid bankruptcy, players sell their
properties for full face value to the debt collector (either another player
or the bank).
These modifications lead to simpler bookkeeping, though they
reduce the realism of the simulation.
Player Strategies
Each player has three main factors to his or her personality:
property buying tactics, trading preferences, and house building strategies.
Correspondingly, each AI personality has three parts, which are subdivided
into base preferences and modifiers. For each of the three parts,
a player has one base preference and a binary value (y/n) for each of the
modifiers.
Property Buying
Base Preferences:
Always buy property when player can
afford it
Only buy property if player will still
have a certain amount of padding left over (i.e., $200)
Randomly (50/50) decide whether of
not to buy property at each opportunity
Modifiers:
Buy property to match other properties
already owned (y/n)
Buy to prevent other players from
making complete sets (y/n)
Property Trading
Base Preferences:
Never trade with other players
Only trade with players that have smaller
assets (cash + face value of properties and houses)
Always willing to trade
Randomly (50/50) decide whether to trade at
each opportunity
Modifiers:
Require that trade be a good deal (property
received worth equal to or more than property given)
Do not help opponents make complete sets
House Building
Base Preferences:
Build as many houses as possible on a particular
set
Spread houses among all sets
Modifier:
Build houses where other players are likely
to land (by rolling a 6, 7, or 8)
Experiment
To test these different personalities, we ran a series of four-player
trials, with one trial consisting of 10,000 games. By running a large
number of iterations, the role of luck was minimized, accounting for approximately
1% error. Matching four identical players, knowing that each
should win 25% of the time, we saw the players win between 24 and 26% of
the games.
Rather than exhaustively testing all combinations of strategies against
each other, we used a form of interactive evaluation, using the results
of previous trials to chose future matches. Each trial generally
focused on the effect of a single trait: a base preference or a modifier.
The trials consisted of three types of matches:
 |
Equal Pairs
Used to compare the effect of a particular trait
on otherwise identical players. Players 0, 1, 2, and 3 have exactly
the same preferences, but players 0 and 1 have the desired trait turned
on, while players 2 and 3 have it turned off. The
players' performance is judged against a 25% level, the average
player's success. |
 |
Unequal Pairs
Used to view the effect of a particular trait
across several player profiles. Players 0 and 1 have identical profiles,
except that player 0 has the desired trait turned on, while player 1 has
it turned off. Players 2 and 3 have identical profiles, though they
might be completely different from the profiles of players 0 and 1, and
again, player 2 has the trait turned on and player 3 has it turned off.
Many sets of unequal pairs are played to reveal the change in magnitude
of a trait's influence. |
 |
Quartets
Used to analyze the relative effects of non-binary
traits, such as base preferences. The participants have identical
profiles except each has a different base preference (i.e. never trade,
always trade, trade if wealthier than opponent, random trade). The
results indicate which players should be paired for future matches. |
Results
Property Buying
Always buy vs. Leave padding (but also buy matches and pieces
opponents want)
| Players that always bought properties performed slightly (3%) worse
than players that left $200 padding in their accounts, dipping below that
level only for matching pieces and pieces opponents needed to make complete
sets. |
 |
|
Effect of changing padding
|
|
|
Key:
Player 0: Always buy
Player 1: Leave padding, but buy matches
Player 2: Leave padding, but buy matches and pieces opponents want
Player 3: Random buy
Changing the padding value dramatically altered the success of particular
strategies. At a relatively benign $200, all players fared about
the same, though the player that always bought properties performed slightly
below the others. With the padding set to $1000, the players that
aimed to keep this amount in their accounts performed much better than
the player that always bought and the random buyer. At $1250, the
results leveled off again, but at a prohibitively high $1500 (the amount
the players received at the start of the game), the players that kept padding
in their accounts began to lose much more often. |
Other property buying results
-
Property buying results largely influenced by other players' trading strategies
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Players that dipped beneath padding threshold to buy matches performed
2.5% better than those that left padding
-
Players that did not buy matches, but did dip beneath the padding threshold
to buy pieces opponents wanted performed 2.5% worse than those that left
padding
-
Players that bought both had average performance
Property Trading
Never trading
| The most dramatic effect of any strategy was the detrimental effect
of never trading with opponents. Players that rigidly refused to
trade with others won only a small (4%) portion of the games. Additionally,
many (6%) of the games would have run infinitely.
Most likely the infinite games were caused by the lack of complete sets.
Since players must own an entire set for development, rents remained negligible
and players continued to lap Go. |
|
Not trading pieces opponents want
| Players that refused to make trades that helped opponents make complete
sets also fared below average. Players that always traded performed
8% better than those that held back properties that opponents wanted.
Perhaps a more complicated, flexible AI strategy could be devised to
retain pieces opponents wanted and still outperform strategies that always
trade pieces. |
|
Other property trading results
-
Requiring trade be a good deal lowered chances of winning by as much as
9%
-
Always trading performed 4% better than trading only if wealthier than
opponent
-
Trading restrictions strongly influenced game length
House Building
Build up one set at a time vs. Spread across board
| Players that completely developed one set before building houses on
another performed 2% better than players that spread houses across all
sets.
Building houses where other players were likely to land gave no appreciable
benefit. |
|
Composite "best" strategy
The following traits increased players' chances of winning slightly.
However, the benefit is too small to notice in only a few games.
So, for a human player, this strategy is not appreciably better.
-
Build houses on one set at a time
-
Build on premium properties (rather than locations where other players
are likely to land)
-
Keep a substantial amount of money (around $1000) in cash unless a property
matches others owned
-
Do not buy properties just because opponents want them
-
Always accept trades
Conclusions
Though there are a few detrimental tactics (i.e., refusing to trade),
there are no exceptionally beneficial strategies for a human playing a
limited number of games. Thus, Monopoly is a highly balanced game.
Future Work
To further explore the game space, we will investigate other avenues,
including:
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Running the "best" strategy against many others
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Testing other strategies: ganging-up, building houses without owning complete
sets, complex auctions
-
Considering other metrics for fun: game length, time between first bankruptcy
and end of game, house building
-
Applying a more systematic evaluation method like competitive evolution
or another optimization technique
Finally, we will move the local problem back to the wider scope of games
in general. We will investigate other, less luck-bound games such
as Risk and chess. Finally, we will apply these principles to the
new massively multi-player server games, hoping to remedy game imbalances
with successful techniques from traditional games.
Participants
Moira Burke
eclipse@gladstone.uoregon.edu
http://gladstone.uoregon.edu/~eclipse
Moira is a senior majoring in Computer
and Information Science at the University
of Oregon. She also participated in the CRA DMP in the summer
of 1999, implementing tools for visualizing
parameters of physical motion.
Jessica Hodgins
jkh@cc.gatech.edu
http://www.gvu.gatech.edu/~jessica.hodgins/
Jessica is an Associate Professor and Assistant Dean in the College
of Computing at Georgia Tech. |
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