Game Theory Lecture: From Beauty Contest to Strategic Form Games

The class plays a guessing game where each student selects a number between 1 and 100. The winner is the person whose guess lies closest to two‑thirds of the class average. An “ultra‑rational” argument predicts that everyone should guess 0, because iterated reasoning drives choices downward. In practice, the winning guess was 12 and the class average settled around 18, showing that real participants deviate from the ultra‑rational prediction. The professor notes that guessing 100 is generally ineffective, and the results shift over repeated rounds as players adjust their expectations.

Individual vs. Interactive Decision Making

Utility theory begins with single‑agent lotteries, where a decision maker evaluates probabilistic outcomes. The lecture expands this framework to multi‑agent settings, defining a game as a decision problem involving several agents. Each player’s best decision depends on the decisions of others, echoing the quote, “The best decision for each of them depends upon the decisions that other players are making.” This shift requires a systematic way to describe who moves, what moves are available, what outcomes result, and what each player knows.

Extensive Form Games

The professor models games as rooted trees, where each node records the history of moves that have occurred. The PAPI acronym—Players, Actions, Payoffs, Information—captures the essential ingredients of any extensive‑form representation. Players are the strategic agents, Actions are the moves available at each node, Payoffs assign von Neumann‑Morgenstern utilities to terminal nodes, and Information specifies what each player observes when making a move. Nature appears as a special, non‑strategic player that selects moves according to fixed probabilities; it has no payoffs or information sets. The lecture emphasizes that utility always refers to von Neumann‑Morgenstern utility, not raw monetary amounts, reinforcing the distinction highlighted in the quote, “There's a crucial distinction between the consequence, which here is measured in dollars, and the utility the player gets from that consequence.”

Information Sets

An information set groups together decision nodes that a player cannot tell apart. The lecture defines an information set as a partition of a player’s decision nodes where the player lacks the ability to distinguish between nodes within the same set. Simultaneous moves are modeled by placing the relevant nodes in a common information set rather than stacking two players at a single node. The professor stresses two restrictions: every node in an information set must offer the same feasible actions, and the root of the game must be a singleton information set.

Strategies

A strategy is a complete contingent plan that specifies an action for every information set a player might encounter. The professor repeats the definition, “A strategy is a complete contingent plan.” Strategies must account for all possible future states, even those that seem unlikely, because a player cannot know which information set will be reached. The number of strategies grows exponentially with the number of information sets; if a player has two moves at each of n information sets, the total number of strategies equals (2^{n}).

Strategic Form Games

To simplify analysis, the lecture shows how to reduce an extensive‑form game to its strategic form. The strategic form lists each player, their strategy sets (S), and payoff functions (u) that map strategy profiles to real numbers. A strategy profile induces a lottery over terminal nodes, allowing the calculation of expected utility using von Neumann‑Morgenstern preferences. The professor summarizes, “The strategic form representation is a way of reducing the game simply to its strategic components.” This reduction enables comparison of strategy profiles without tracing the full tree.