18.1 Properties of Multiagent Environments
Agents may have aligned or conflicting goals, complete or partial information, and simultaneous or sequential actions. These properties shape the appropriate model.
AI · STRATEGIC INTERACTION
18 · Multiagent Decision Making
When several agents act, outcomes depend on their joint choices. Game theory analyzes incentives, cooperation, and the rules used to make collective choices.
18.2 Non-Cooperative Game Theory
Each agent chooses a strategy to maximize its own payoff. A Nash equilibrium is a strategy profile where no agent benefits from changing unilaterally.
18.3 Cooperative Game Theory
Cooperative models study coalitions, how their joint value is formed, and how gains can be fairly or stably allocated among members.
18.4 Making Collective Decisions
Voting rules aggregate individual preferences into a group choice; different rules can produce different outcomes, so fairness properties matter.
Key terms and payoff trace
- Strategy profile
- The complete choice of each player; a matrix cell gives the resulting payoff pair (row player first).
- Best response
- A strategy that maximizes one player's payoff while holding the other player's choice fixed.
- Nash equilibrium
- A profile where every player's strategy is a best response to the others; no unilateral deviation improves payoff.
- Coordination trade-off
- Both matching outcomes are pure equilibria here. Alpha/Alpha gives 7 total points and Beta/Beta also gives 7; which equilibrium is selected depends on expectations.
Trace: if the column player chooses Beta, the row player compares 0 from Alpha with 3 from Beta, so Beta is the best response. If both choose Beta, the column player likewise prefers Beta (4 over 0). At mismatched choices, a player can improve by switching to match. The payoffs are illustrative and do not imply that cooperation or any strategy is universally optimal.
Interactive payoff matrix
Coordinate on a shared project
Both agents prefer coordinating on either option, but each gets a higher payoff from their own favorite. Payoffs are (row player, column player).
| Column: Alpha | Column: Beta | |
|---|---|---|
| Row: Alpha | 4, 3 | 0, 0 |
| Row: Beta | 0, 0 | 3, 4 |
Pure Nash equilibria: Alpha/Alpha and Beta/Beta.
Select a strategy profile.
Demo performance
For this fixed 2×2 matrix, updating the selected profile, highlighting the four cells, and choosing the row player's best response each take O(1) time and O(1) auxiliary space. The payoff table is already in the DOM. More generally, storing an m×n payoff matrix takes O(mn) space; scanning a fixed column for a row best response takes O(m) time, while enumerating every profile takes O(mn).