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How do you teach and learn subgame perfect equilibrium concepts and techniques effectively?

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Subgame perfect equilibrium (SPE) is a refinement of the Nash equilibrium concept that applies to sequential games. It requires that players choose optimal strategies not only for the whole game, but also for every subgame that may arise. Learning and teaching SPE can be challenging, as it involves backward induction, extensive form representation, and subgame identification. Here are some tips and techniques that can help you master this important concept in game theory.

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Dr Djamila Amimer

Helping Businesses Unlock AI Potential | CEO & Founder | Top 10 Global Thought Leaders on AI, Predictive Analysis and…

1 What is a subgame?

A subgame is a part of a game that starts at a specific node and includes all the possible moves and outcomes that follow from that node. A subgame must be self-contained, meaning that it does not depend on the previous history of the game. For example, in the ultimatum game, where one player proposes how to split a sum of money and the other player accepts or rejects the offer, there is only one subgame: the whole game itself. However, in a game of chess, there are many subgames that start at different positions on the board.

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2 How to use backward induction?

Backward induction is a method of finding SPE by working backwards from the end of the game. It involves identifying the optimal choices for each player at every decision node, starting from the last ones and moving up to the first one. The idea is that rational players anticipate the future actions and payoffs of their opponents and themselves, and choose the best response accordingly. For example, in the centipede game, where two players take turns choosing between taking a larger share of a growing pot or passing it to the next player, backward induction shows that the SPE is to take the pot at the first opportunity.

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Dr Djamila Amimer

Helping Businesses Unlock AI Potential | CEO & Founder | Top 10 Global Thought Leaders on AI, Predictive Analysis and Emerging Tech. | AI Speaker | AI Author| LinkedIn Top AI Voice
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Worth pointing out that backward induction cannot be used in games where we have imperfect or incomplete information as it results in cutting through non singleton information sets.

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3 How to draw extensive form?

Extensive form is a graphical representation of a game that displays the sequence of moves, the information available to each player, and the payoffs for each outcome. It is beneficial for applying backward induction and recognizing subgames. When drawing an extensive form, you must adhere to certain guidelines: use a tree structure with branches for each move, label each node with the player who makes the move, utilize dotted lines to indicate information sets where a player does not know which node they are at, write the payoffs for each player at the end of each branch, and employ parentheses to separate the payoffs for different players.

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4 How to find subgames?

To identify subgames in an extensive form, you must locate nodes that initiate a game that is independent of the prior history. These subgames must contain all the nodes and branches originating from a single node, without cutting across any information sets or including nodes that are part of another subgame. For instance, in the game of chicken, where two drivers decide whether to swerve or stay on a collision course, there are four subgames - one for each potential combination of moves.

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Madison Laprise

Cognitive Science at Yale | Graduating 2025 | Aspiring Data Analyst
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A subgame of a perfect information game is part of a tree that: a) starts from a singleton node and; b) contains all successor nodes Find the smallest subgame first, circling each subgame as you move back through the tree

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5 How to check for SPE?

To check if a strategy profile is a SPE, you need to verify that it is a Nash equilibrium for every subgame of the game. This means that no player can benefit from deviating from their strategy in any subgame, given that the other players do not deviate. You can use backward induction to find the best response for each player at each node, and compare it with their strategy. If they match, then the strategy is a SPE. If not, then the strategy is not a SPE.

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6 How to practice SPE?

One way to practice SPE is to solve exercises that involve finding or checking for SPE in different games. You can use online resources, textbooks, or courses that provide examples and solutions of SPE problems. Another way to practice SPE is to play games that require SPE reasoning, such as chess, poker, or tic-tac-toe. You can challenge yourself or others to find the optimal moves and outcomes in these games, and explain your reasoning using backward induction and subgame identification.

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7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How do you teach and learn subgame perfect equilibrium concepts and techniques effectively?

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1 What is a subgame?

2 How to use backward induction?

3 How to draw extensive form?

4 How to find subgames?

5 How to check for SPE?

6 How to practice SPE?

7 Here’s what else to consider

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