Effect of information on collusion strategies in single winner multi agent games
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Effect of Information on Collusion Strategies in Single winner, multi-agent games. December 2, 2010 Nick Gramsky Ken Knudsen. Contents. 1. Motivation 2. Identification of Collusion 3. Classification of Coalitions 4. Implementation 5. Results 6. Conclusions. Motivation.

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Effect of Information on Collusion Strategies in Single winner, multi-agent games

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Effect of Information on Collusion Strategies inSingle winner, multi-agent games

December 2, 2010

Nick Gramsky

Ken Knudsen


Contents

  • 1. Motivation

  • 2. Identification of Collusion

  • 3. Classification of Coalitions

  • 4. Implementation

  • 5. Results

  • 6. Conclusions


Motivation

  • Reasons to Collude

    • Improve position relative to other agent(s)

    • Self-preservation / Survival

  • Explicit Collusions

    • Alliances

    • Survival

    • Truces

  • Implicit Collusions

    • Minimax against strongest player

    • Tit-for-tat


Contents

  • 1. Motivation

  • 2. Identification of Collusion

  • 3. Classification of Coalitions

  • 4. Implementation

  • 5. Results

  • 6. Conclusions


Identification

  • Find course grained collusive behavior

  • 1.Offensive-based collusion

    • Multiple agents attacking a single agent for a fixed number of rounds

    • In our examples, we limited this to 1 round.

  • 2. Defensive-based collusion

    • Multiple agents not attacking each other over a fixed number of rounds.

    • In our examples, we limited this to 2 rounds.


IdentificationOffensive based coalitions


IdentificationDefensive based coalitions


Contents

  • 1. Motivation

  • 2. Identification of Collusion

  • 3. Classification of Coalitions

  • 4. Implementation

  • 5. Results

  • 6. Conclusions


Classification Offensive based behaviors

  • Socially inclined behavior

    • For some predefined time, if target satisfies the following, then we define the actions of the attacking players as being 'socially oriented‘

    • h(x) is a heuristic function for any adversary.

      • vh(x) when dealing with different layers of fog

  • Else: Some other collusive behavior


Classification Offensive based algorithm


Classification Defensive based algorithm


ClassificationMissed opportunities 

  • Classify a missed opportunity by finding players that:

    • for a predefined period were not attacked above a certain percentage and…

    • satisfy either their power heuristic or visual heuristic (below) threshold


Contents

  • 1. Motivation

  • 2. Identification of Collusion

  • 3. Classification of Coalitions

  • 4. Implementation

  • 5. Results

  • 6. Conclusions


Implementation

  • Used Warfish to play games of Risk.

    • Free website warfish.net

  • Risk is a zero-sum game where players seek (simulated) world domination! 

  • Only one winner, the last remaining contestant.

  • Attacks are made via dice (random number generator)

  • Amass armies, grow in power, rule the world!

    • Or at least the world represented on a board...


ImplementationEnvironment

  • Reduced resource strategies

  • Randomized players

  • Set card trade-in values to be constant (5)

  • Disabled card capture on elimination

  • Multiple map types

    • Larger than original Risk board

    • Reduces board specific strategies in analysis


ImplementationWorld Map


ImplementationEurope Map


ImplementationFog of War

  • Varied amount of information available to all agents via different levels of 'fog of war'.

  • 6 different levels of fog available in game

    • Level 0: No fog (perfect information)

    • Level 1: See all occupations, neighboring units only

    • Level 2: See all occupations (no units)

    • Level 3: Only see neighboring occupations and units

    • Level 4: See only neighboring occupations

    • Level 5: Complete fog (only know about self)

  • Tested with 3 levels of fog

    • {0,1,3}


ImplementationOracles

  • Participants who annotated their strategies and behaviors as games were played

  • Compared oracle annotations to game data

    • Spot-check that analysis found collusion

    • Though noisy, analysis and annotations were inline with game history.


Contents

  • 1. Motivation

  • 2. Identification of Collusion

  • 3. Classification of Coalitions

  • 4. Implementation

  • 5. Results

  • 6. Conclusions


ResultsCollusion vs Game length

x-axis: Number of turns

y-axis: Number of "interesting" windows

θh = 1.3 per 1 turn window


ResultsOffensive

  • Players all gang up on Yellow.

  • Validated by Oracle annotations.

  • Game: 98478150

  • Map: World

  • Fog Level: 1


ResultsOffensive

  • Minmax against Blue

  • Confirmed by reading through the transcript.

    • Blue quickly gained power

    • Challenged remaining players to team up against him

  • Game: 97976903

  • Map: Europe

  • Fog Level: 0

“Right now (Yellow) knows that if he does not get both you (Red) and (Green) on his side, this game will be won by me”


ResultsOffensive

Games 98478150 (left) and 97976903 (right)

x-axis: Number of turns

y-axis: Number of "interesting" windows

θh = 1.3 / 1 turn window


ResultsOffensive & Defensive

  • Minimax against strongest player

  • Towards the end of the game, explicit truce between top 2 players

  • Game: 12069561

  • Map: Europe

  • Fog Level: 0


ResultsDefensive

*Game:

12069561

Scatter plot of number of windows classified as defensive-oriented for all games.

x-axis: number of turns

y-axis: number of interesting windows

θ = 0.05


ResultsOracle

  • Oracle self-interest annotations (Blue)

  • Game: 88318444

  • Map: World

  • Fog Level: 1

x-axis: Number of turns

y-axis: Number of "interesting" windows

θh = 1.3 / 1 turn window


ResultsFog Level 3

  • Typical of the layer 3 games.

  • Everything breaks down. Players can’t figure out who is in the lead until it is too late.

  • Game: 67785982

  • Map: Europe

  • Fog Level: 3


Results

  • Collusion % is percentage of available windows where remaining players direct more than 75% of attacks towards target.

  • Social % is percentage of available windows with same criteria as above BUT the target satisfies heuristic thresholds from earlier

  • θh = 1.3 / 1 turn window

  • Target’s residual power

    • 43.3% (4-player)

    • 65% (3 player)

  • θh = 1.6 / 1 turn window

  • Target’s residual power

    • 53.3% (4-player)

    • 80% (3 player)


ResultsEurope Map

  • θh = 1.3

  • θh = 1.6


ResultsWorld Map

  • θh = 1.3

  • θh = 1.6


Contents

  • 1. Motivation

  • 2. Identification of Collusion

  • 3. Classification of Coalitions

  • 4. Implementation

  • 5. Results

  • 6. Conclusions


Conclusions

  • Presented a basic algorithm to identify and classify collusion

  • Games with unusually large number of collusive behaviors tended to prolong games beyond the average.

  • As fog increased (information decreased), collusive behaviors diminished.

  • Results were consistent across maps.

  • Level 0 data was consistent between our volunteers and the public.

  • Analysis supported by Oracle annotations and in-game conversations.


Conclusions

  • Visual heuristic does not hold well for fog games

    • Based on a knowledge of territories and bonuses

  • Limited data sets

    • Time limitation

      • Short time-frame for project

      • Games averaged 20 days to complete

    • Require more experiments with fog levels

  • Data integrity

    • Games had large variance in player abilities

    • Players were involved in multiple simultaneous games

      • May have forgotten strategy

      • Players may have a predefined disposition towards other players (Social Value Orientation)


ConclusionsFuture Work

  • Investigate possible equilibrium in collusions versus game length.

  • Lag response for social orientation.

    • Once the strongest player is removed from power, it can take a few rounds for the coalition to change strategies.

  • As information decreases, agents tend to collude less.  Why?

    • fairness

    • poor assessment of board

  • Mix socially oriented bots with human players


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