Chapter 2: Representing Strategic Interactions#

Last chapter, we looked at a single decision-maker facing uncertainty. Now we turn to games where your outcome depends on what others do, not just your own choice. To think about these situations, we need a clear way to describe who moves when, what they know, and what they care about. This chapter gives you that language.

The Big Picture#

Every game, from chess to a price war between companies, has the same basic parts: players, actions, information, and payoffs. Our job is to write down those parts in a clear, precise way. We’ll meet two main formats: the extensive form (a game tree) and the strategic form (a payoff matrix). We’ll see how they connect. Along the way we’ll handle situations where players don’t know everything, random moves by Nature, and subgames. By the end, you’ll be able to take a real strategic situation and turn it into a formal game we can solve later.

The Game Tree: Extensive-Form Games#

The easiest way to show a sequential game is to draw a tree. A game tree is a picture of the game’s moves, like a branching diagram. The tree starts at the root (the beginning of the game). From there, branches show the possible choices. Each point where a player has to decide is a decision node. At the end of each branch, we reach a terminal node (a leaf) where the game ends. At each terminal node we write the players’ payoffs, which are numbers that tell us how happy they are with that outcome. For example, means Player 1 gets 3 and Player 2 gets 1.

Let’s look at a tiny example. Player 1 moves first. She can pick Left or Right. If she picks Left, then Player 2 gets to choose Up or Down. If she picks Right, the game ends right away, and they get payoffs . The tree has a root (P1), two branches from the root, one leading to a P2 node and one leading to a terminal node, and so on.

Extensive-form game: A game tree with: a root, decision nodes labelled with the player who moves, actions (choices) at each node, information sets (groups of nodes that look the same to a player), and payoffs at the terminal nodes.

If every decision node is in its own separate information set, the game has perfect information. That means when it’s your turn, you know exactly where you are in the tree. Chess is like this: you can always see the whole board. Later we’ll see what happens when you don’t know exactly where you are.

📝 Section Recap: A game tree shows who moves when and what happens. The extensive form is the blueprint of the game, including the order of moves and what players know at each step.

Information Sets and What Players Know#

In many real games, you don’t know exactly what happened before you move. A card player can’t see the other player’s hand. A company doesn’t know its rival’s costs. To model this, we group decision nodes into information sets.

An information set is a group of decision nodes that look the same to the player. The player knows they are in that group, but not exactly which node. All nodes in the group must belong to the same player and have the same choices available. (If the choices were different, the player could figure out which node they’re at.) We draw a dashed line to connect the nodes in the same information set.

If every information set has just one node, the game has perfect information. The player always knows the exact history. If some information set has more than one node, the game has imperfect information. The classic example is Matching Pennies: Player 1 secretly picks Heads or Tails. Player 2, without knowing the choice, then picks Heads or Tails. In the tree, P2’s two decision nodes are joined into one information set.

Perfect information game: An extensive-form game in which every information set is a singleton (just one node).

📝 Section Recap: Information sets tell us what a player knows. If every set has just one node, the player knows everything. If some sets have more than one node, the player is uncertain — and that’s what makes many games interesting.

Chance Moves: Nature’s Role#

Not all uncertainty comes from other players’ hidden actions. Sometimes the world itself is random — a die roll, a weather shock, a random draw from a deck. We capture this by introducing a special player called Nature (or Chance).

Nature is a non-strategic player. She picks moves randomly, with given probabilities. In the tree, a Nature node has branches labelled with probabilities that add up to 1. Everyone sees the result, unless we hide it using information sets.

For example, imagine a simple card game: Nature first deals a high card (probability ) or a low card (probability ). Player 1 sees the card and can then bet or fold; Player 2 sees only whether Player 1 bet, not the card itself. The tree has a Nature root, then a P1 decision node (two nodes, one for each card, but they are in separate information sets because P1 sees the card), then P2 decision nodes grouped in information sets that hide the card.

Nature lets us include any external randomness cleanly. Later, when we compute expected payoffs, we’ll average over Nature’s probabilities.

📝 Section Recap: Chance moves are modelled by a non‑strategic player, Nature, whose actions follow known probabilities. They let us include real‑world randomness without complicating the strategic logic.

Strategies: Complete Plans of Action#

A player doesn’t just make a single move; they need a plan that says what to do in every situation they might face. We call such a plan a pure strategy.

Pure strategy: A complete plan that picks one action for each of the player’s information sets.

Think of it as a full instruction manual: “If I find myself at information set A, I will do this; if at information set B, I will do that.” The plan must cover every information set the player has, even those that may never be reached if the player follows the plan — because the plan has to be ready for any possible path the game might take.

For instance, suppose Player 1 has two information sets. At the first she can choose or , and at the second she can choose or . Then her possible pure strategies are all combinations: , , , — four strategies in total. In general, if a player has information sets and actions at information set , the number of pure strategies is .

A strategy is called pure to distinguish it from a mixed strategy, where the player randomises over actions. We’ll explore mixed strategies later; for now, a pure strategy is a deterministic plan.

📝 Section Recap: A pure strategy is a full contingency plan — one action for every information set. Counting strategies as combinations of actions is the first step in moving from the tree to a compact table.

From Trees to Tables: The Strategic Form#

Once we have listed every player’s set of pure strategies, we can compress the whole game into a matrix. This is the strategic form (also called the normal form).

Imagine a two-player game. Let Player 1’s pure strategies be and Player 2’s pure strategies be . For each pair we can trace through the tree: the strategies dictate a unique path from the root to a terminal node, and we read off the payoff vector. We then write that vector in the cell where the row for meets the column for .

A simple example: Player 1 chooses or ; Player 2, without knowing P1’s choice, chooses or . The tree has P1’s root, then two P2 nodes in a single information set. P1’s strategies are just and . P2’s strategies are and (only one information set). Suppose the payoffs are:

The strategic form is a matrix:

When players move simultaneously, we can still use a tree: we let Player 1 move first, then place all of Player 2’s nodes in a single information set. Because P2 doesn’t see P1’s move, it’s as if they move at the same time. The strategic form of that tree is exactly the simultaneous‑move matrix we’re used to. So the extensive form with imperfect information is the standard way to represent simultaneous moves inside a tree.

Sometimes different pure strategies lead to exactly the same outcome for every possible choice of the others. We can then simplify the strategic form by keeping only one representative of each equivalent set of strategies. This gives the reduced strategic form, which removes redundant rows or columns without losing any strategic information. For example, if two of Player 1’s strategies yield identical payoffs against every strategy of Player 2, we can drop one.

📝 Section Recap: The strategic form collapses the whole tree into a payoff matrix by considering all possible pure‑strategy profiles. It is the workhorse for equilibrium analysis, and we can always obtain it from an extensive‑form game.

Subgames: Games Within Games#

Sometimes a part of a game tree looks like a complete game in itself. We call such a part a subgame.

Subgame: A piece of the tree that (i) starts at a node that is alone in its information set (so the player knows exactly where they are), (ii) includes everything that follows that node, and (iii) does not break any information set: if a node is in the subgame, all nodes in the same information set must also be in the subgame.

Condition (i) guarantees that the player at the start of the subgame knows exactly that they are entering that subgame. Condition (iii) ensures that no player is forced to have partial information about what happened inside the subgame versus outside it.

In a game of perfect information, every decision node starts a subgame (the subtree rooted at that node). In a game with imperfect information, many nodes will not start subgames because they belong to non‑singleton information sets. For instance, in the simultaneous‑move tree we built earlier, the P2 nodes are not singletons, so they do not start subgames; the only subgame is the whole game itself.

📝 Section Recap: A subgame is a self‑contained subtree that respects information sets. It is a crucial concept for refining our predictions in dynamic games.

Cooperative vs. Noncooperative Game Theory#

All the representations we have built — extensive form, strategic form, subgames — belong to noncooperative game theory. In noncooperative games, we focus on individual players who choose their own strategies to maximise their own payoffs. They cannot sign binding contracts; any cooperation must emerge from self‑interest.

Cooperative game theory takes a different view. It assumes that players can form groups (coalitions) and make binding agreements about how to share the total payoff. The main object is a characteristic function that says how much payoff each coalition can guarantee for itself. The question then becomes: how should the coalition’s worth be divided among its members in a stable way? Ideas like the core and the Shapley value live here.

Why mention this now? Because the same real situation can often be modelled in either way, and the choice of framework depends on whether binding agreements are feasible. In this course we will spend most of our time in the noncooperative realm, but the cooperative perspective will reappear when we study bargaining and coalitional games. For the rest of this chapter and the ones that immediately follow, we are firmly in the noncooperative world — trees, matrices, and individual strategies.

📝 Section Recap: Noncooperative theory models individual strategic choice without binding agreements, using extensive and strategic forms. Cooperative theory assumes coalitions can form and focuses on dividing joint payoffs. The two approaches complement each other.

Summary#

Now you have a toolkit for writing down any strategic situation. The extensive form shows the order of moves, what players know, and where chance appears. Information sets handle hidden moves and simultaneous choices. Pure strategies give you complete plans, and the strategic form turns those plans into a payoff matrix. Subgames let you spot self‑contained parts of the game, and the cooperative vs. noncooperative distinction helps you choose the right model. With these tools, you’re ready to predict how rational players will behave.