# Move Sequences and Distributions

Two tools at our disposal for understanding the dynamics inherent in a given power structure are move sequences and move distributions.

## Move Sequences

Move sequences show how a power structure changes over time, with attention drawn to the active agents. Sometimes move sequences are created by hand, such as to show the progression of an actual historical event. But they can also be generated by simulations in which each move is based on game tree search. In these sequences, the sizes of the agents are typically normalized at each time step such that the largest agent always has a size of 1, and the moving or "focal" agent is highlighted in red.

### Sequential Move Simulations

In a simulation based on sequential moves, the agents move one at a time and the one taking the action is circled in red: The behavior that emerges from sequential move simulations is very sensitive to the order in which the agents move. Unless otherwise indicated, we use random move orders in which the agents have an equal probability of moving. However, when generating the game tree to determine each individual move, an agent's probability of moving is proportional to their PrinceRank. This gives every agent the equal opportunity to move, but when deciding what their move should be, they pay extra attention to the more important agents.

### Simultaneous Move Simulations

In a simulation in which agents move simultaneously, the agents highlighted in red are the ones whose actions caused the power structure to change from the last time step. In the sequence above, the agents that are not highlighted did in fact move, they just either did not change their tactic or their move was made irrelevant by another agent whose tactic was more negative. It is generally possible to figure out which agent caused which changes, but there are situations when simultaneous move sequences are ambiguous.

## Move Distributions

A second way to explore the model is to look at the best moves of a given player, in a given scenario, over a variety of random move orders. These "move distributions" are presented as follows: The initial state is shown on the left, at t=0. On the right are the likely moves of the focal player, with their associated probabilities based on the move orders that were randomly sampled. Move distributions estimate the strength of a player's move options.

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