Question

Consider the following strategic setting involving a cat named Baker, a mouse named Cheezy, and a dog named Spike. Baker's objective is to catch Cheezy while avoiding Spike; Cheezy wants to tease Baker but avoid getting caught; Spike wants to rest and is unhappy when he is disturbed. In the morning, Baker and Cheezy simultaneously decide what activity to engage in. Baker can either nap \((\mathrm{N})\) or hunt \((\mathrm{H})\), where hunting involves moving Spike's bone. Cheezy can either hide (h) or play (p). If nap and hide are chosen, then the game ends. The game also will end immediately if hunt and play are chosen, in which case Baker captures Cheezy. On the other hand, if nap and play are chosen, then Cheezy observes that Baker is napping and must decide whether to move Spike's bone \((\mathrm{m})\) or not \((\mathrm{n})\). If he chooses to not move the bone, then the game ends. Finally, in the event that Spike's bone was moved (either by Baker choosing to hunt or by Cheezy moving it later), then Spike learns that his bone was moved but does not observe who moved it; in this contingency, Spike must choose whether to punish Baker (B) or punish Cheezy (J). After Spike moves, the game ends. In this game, how many information sets are there for Cheezy? How many strategy profiles are there in this game?

Short answer

Cheezy has 2 information sets and there are 16 strategy profiles in total.

Step-by-step solution

Understanding Information Sets

In game theory, an information set for a player encompasses all the decision points where the player has the same information regarding what has occurred so far in the game. Cheezy has two decision points: 1) In the initial action where he chooses between hiding (h) or playing (p), and 2) if the combination "nap" and "play" are chosen, he will then decide whether to move Spike's bone (m) or not (n). In the first instance (h or p), he has complete information about the game; in the second decision, it only occurs when the combination (N, p) happens. So, Cheezy has separate information sets for each decision point, totaling two distinct information sets.

Counting Strategy Profiles

A strategy profile in a game is a specification of strategies for each player, detailing what each player will do during their turn. In this game, Baker has 2 choices: "Nap (N)" or "Hunt (H)". Cheezy initially has 2 choices: "Hide (h)" or "Play (p)". If Cheezy plays, and Baker naps, Cheezy will then decide on moving Spike’s bone: either move (m) or not move (n). Spike only acts if the bone was moved, choosing between punishing Baker (B) or Cheezy (J). Thus, Baker has 2 strategies, Cheezy has 4 strategies (h, p & m, n), and Spike also has 2 strategies. Therefore, the total number of strategy profiles is calculated by multiplying the number of choices for each player: \(2 \times 4 \times 2 = 16\).

Key concepts

Information Sets

In game theory, an information set refers to the collection of all possible decisions a player might make at a given point in a game, based on the information available to them. For Cheezy, the mouse, identifying information sets involves understanding his decision points within the game's structure. His goal is to navigate the game without getting caught by Baker.
Initially, Cheezy faces a decision between hiding (h) or playing (p). At this point, he knows everything about the game's setup but nothing about Baker's move since they act simultaneously. Thus, this forms his first information set.
If the game progresses with Baker choosing to nap (N) while Cheezy plays (p), Cheezy faces a second decision: whether to move Spike's bone (m) or refrain from moving (n). In this case, Cheezy is aware that Baker is napping, which provides a distinct context compared to his initial choice, so it forms a second information set. Consequently, Cheezy has two separate information sets in this game.

Strategy Profiles

Strategy profiles in game theory represent a complete plan for each player, detailing the actions they will take throughout the game. Each player selects their strategies simultaneously, and a strategy profile captures all players’ decisions into a cohesive plan.
Baker, the cat, chooses between taking a nap (N) and hunting (H), laying the groundwork for Cheezy's decisions. Initially, Cheezy decides to hide (h) or play (p). If Cheezy moves to playing and Baker naps, Cheezy can choose to move Spike's bone (m) or not (n).
Spike, the dog, only acts if the bone has been moved, and can decide to punish either Baker (B) or Cheezy (J). These decisions together form potential strategy profiles for the game. For Baker, there are 2 strategies (N or H). For Cheezy, there are 4 strategies since his actions depend on Baker's choice, and Spike has 2 options when the bone is moved. Overall, the game presents us with 16 possible strategy profiles, calculated by multiplying the available choices: \(2 \times 4 \times 2 = 16\).

Decision Making

Decision making is a central theme in game theory, with players trying to make the best choices given their goals and the strategies of others. In this scenario, Baker, Cheezy, and Spike all have unique objectives that guide their decision making.
- Baker wants to catch Cheezy, but must avoid disturbing Spike. He decides between napping (N) or hunting (H).
- Cheezy seeks to tease Baker without getting caught, choosing whether to hide (h) or play (p). If Baker naps and Cheezy plays, Cheezy must decide if moving the bone (m) is worth the risk.
- Spike prefers uninterrupted rest and faces the decision of whom to punish (Baker or Cheezy) if his bone is moved.
Each decision is influenced by the potential reactions and strategies of the other players, making strategic thinking and anticipation crucial in shaping the outcome of the game.

Simultaneous Moves

Simultaneous moves are a defining feature of the game involving Baker, Cheezy, and Spike, emphasizing the uncertainty and strategic depth such games possess. In simultaneous move games, players make their choices without knowing the others’ decisions at the same time.
Initially, Baker and Cheezy act simultaneously—Baker chooses between nap (N) and hunt (H), while Cheezy decides to hide (h) or play (p). This aspect of the game creates uncertainty, as each player must anticipate the possible strategy of the other while making their own move.
When both Baker and Cheezy move at the same time, the unpredictability simulates real-life strategic settings where decision makers operate with limited information about others' actions. Successfully navigating this requires careful consideration of all possible outcomes and potential strategies, underscoring the complexity and intrigue typical of game theoretical analyses.