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 Explicit Collusions Alliances Survival Truces Implicit Collusions Minimax against strongest player Tit-for-tat Reasons to Collude Improve position relative to other agent(s) Self-preservation / Survival

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.

Identification Offensive based coalitions

Identification Defensive based coalitions

Contents 1. Motivation 2. Identification of Collusion 3. Classification of Coalitions 4. Implementation 5. Results 6. Conclusions

1. 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 2. Else: Some other collusive behavior Classification Offensive based behaviors

Classification Offensive based algorithm

Classification Defensive based algorithm

Classification Missed 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...

Implementation Environment 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

Implementation World Map

Implementation Europe Map

Implementation Fog 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}

Implementation Oracles 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

Results Collusion vs Game length x-axis: Number of turns y-axis: Number of "interesting" windows θ h = 1.3 per 1 turn window

Results Offensive 1. Players all gang up on Yellow. 2. Validated by Oracle annotations. Game: Map: World Fog Level: 1

Results Offensive 1. Minmax against Blue 2. Confirmed by reading through the transcript. 1. Blue quickly gained power 2. Challenged remaining players to team up against him Game: 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”

Results Offensive x-axis: Number of turns y-axis: Number of "interesting" windows θ h = 1.3 / 1 turn window Games (left) and (right)

Results Offensive & Defensive 1. Minimax against strongest player 2. Towards the end of the game, explicit truce between top 2 players Game: Map: Europe Fog Level: 0

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 *Game: Results Defensive

Results Oracle 1. Oracle self-interest annotations (Blue) Game: Map: World Fog Level: 1 x-axis: Number of turns y-axis: Number of "interesting" windows θ h = 1.3 / 1 turn window

Results Fog Level 3 1. Typical of the layer 3 games. 2. Everything breaks down. Players can’t figure out who is in the lead until it is too late. Game: 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 θ h = 1.3 / 1 turn window Target’s residual power 43.3% (4-player) 65% (3 player) θ h = 1.6 / 1 turn window Target’s residual power 53.3% (4-player) 80% (3 player)

Results Europe Map θ h = 1.3 θ h = 1.6

Results World Map θ h = 1.3 θ h = 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)

Conclusions Future 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