Emergence of good conduct, scaling and zipf laws in human behavioral sequences in an online world

Abstract

We study behavioral action sequences of players in a massive multiplayer online game. In their virtual life players use eight basic actions which allow them to interact with each other. These actions are communication, trade, establishing or breaking friendships and enmities, attack, and punishment. We measure the probabilities for these actions conditional on previous taken and received actions and find a dramatic increase of negative behavior immediately after receiving negative actions. Similarly, positive behavior is intensified by receiving positive actions. We observe a tendency towards antipersistence in communication sequences. Classifying actions as positive (good) and negative (bad) allows us to define binary 'world lines' of lives of individuals. Positive and negative actions are persistent and occur in clusters, indicated by large scaling exponents α ~ 0.87 of the mean square displacement of the world lines. For all eight action types we find strong signs for high levels of repetitiveness, especially for negative actions. We partition behavioral sequences into segments of length n (behavioral 'words' and 'motifs') and study their statistical properties. We find two approximate power laws in the word ranking distribution, one with an exponent of κ ~ -1 for the ranks up to 100, and another with a lower exponent for higher ranks. The Shannon n-tuple redundancy yields large values and increases in terms of word length, further underscoring the non-trivial statistical properties of behavioral sequences. On the collective, societal level the timeseries of particular actions per day can be understood by a simple mean-reverting log-normal model.

Figure 1. Short segment of action sequences of three players, , , and .

(a). Some actions of players 146 and 701 are directed toward player 199. This results in a sequence of received-actions for player 199, . The combined sequence of actions (originated from - and directed to) player 199, , is shown in the last line; red letters mark actions from others directed to player 199. (b) Schematic illustration showing the definition of a binary walk in ‘good-bad’ action space (good-bad ‘world line’). A positive action (C, T, F or X) means an upward move, a negative action (A, B, D and E) is a downward move. Good people have rising world-lines.

Figure 2. Timeseries of the daily number of (a) trades, (b) attacks, (c) communications in the first 1238 days in the game.

Clearly a mean reverting tendency of three processes can be seen. (d) Simulation of a model timeseries, Eq. (1), with . We use the values from the timeseries, , and standard deviation . Compare with the actual in (c). The free parameter in the model is . Parameters are from Tab. 1. Mean reversion and log-normality motivate the model presented in Eq. (1). (e) The distributions of log-increments of the processes and the model. All follow approximate Gaussian distribution functions.

Figure 3. Transition probabilities for actions (and received-actions) at a time , given that a specific action was executed or received in the previous time-step .

(a). Received-actions are indicated by a subscript . Normalization is such that rows add up to one. The large values in the diagonal signal that human actions are highly clustered or repetitive. Large values for and reveal that communication is a tendentially anti-persistent activity – it is more likely to receive a message after one sent a message and vice versa, than to send or to receive two consecutive messages. (b) The ratio , shows the influence of an action at a previous time-step on a following action at a time , where and can be positive or negative actions, executed or received (received actions are indicated by the subscript ). In brackets, we report the Z-score (significance in number of standard deviations) in respect to a sample of 100 randomized versions of the dataset. The cases for which the transition probability is significantly higher (lower) than expected in uncorrelated sequences are highlighted in red (green). Receiving a positive action after performing a positive action is highly over-represented, and vice versa. Performing (receiving) a negative action after performing (receiving) another negative one is also highly over-represented. Performing a negative action has no influence on receiving a negative action next. All other combinations are strongly under-represented, for example after performing a negative action it is very unlikely to perform a positive action with respect to the uncorrelated case.

Figure 4. World lines of good-bad action random walks of the 1,758 most active players (a), distribution of their slopes (b), and of their scaling exponents (c).

By definition, players who perform more good (bad) than bad (good) actions have the endpoints of their world lines above (below) 0 in (a) and only fall into the () category in (b). (d) World lines of action-received random walks, (e) distribution of their slopes and (f) of their scaling exponents . The inset in (d) shows only the world lines of bad players. These players are typically dominant, i.e. they perform significantly more actions than they receive. In total the players perform many more good than bad actions and are strongly persistent with good as well as with bad behavior, see (c), i.e. actions of the same type are likely to be repeated.

Figure 5. Rank ordered probability distribution of 1 to 6 letter words.

Slopes of and are indicated for reference. The inset shows the Shannon -tuple redundancy as a function of word length .