# Backward Induction

Published on :

21 Aug, 2024

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Dheeraj Vaidya

## What Is Backward Induction?

Backward Induction refers to a decision-making method that begins at the end of a problem and reasons backward to know the optimal options at every step. It is useful in pricing derivatives and options, assessing investment decisions, modeling strategic actions in financial markets, and gauging optimal auction strategies.

It identifies the best strategy, analyzes optimal choices in financial models, and forecasts results in competitive situations for a sequential administrative scenario. Economics, finance, and game theory widely use it in solving related problems. It helps make well-informed and correct decisions in complex financial circumstances.

• Backward induction is a decision-making technique that starts at a problem's conclusion and works backward to determine the best options at each stage.
• It tells the most effective strategy, evaluates optimal decisions in financial models, and predicts outcomes in competitive scenarios within a sequential administrative setting.
• The process involves identifying the final decision point, analyzing optimal choices, working backward, considering future choices, determining the optimal choice for the current point, repeating steps, and revealing the best strategy for the entire sequence.
• It employs reasoning from the end to the start, while forward induction advocates reasoning from the start to the end.

### How Does Backward Induction Valuation Work?

Backward induction in economics is defined as a valuation method to determine the fair value of an asset by beginning from its maturity and working backward to its current value. To do so, it starts from the ending nodes of a binomial interest rate tree to earlier nodes by determining the asset value based on the final node value. Moreover, the process considers the average current asset value of the two probable future interest rates. Nevertheless, its assumptions, like perfect details on future interest rates and asset values, besides decisions of retail investors to maximize expected utility, may sometimes be unrealistic.

In finance, a binomial tree determined using a binomial model helps apply binomial inductions on bond valuations. Its starting point is the final node of the receipt of the principal by investors and final coupon redemption. Next, the value of the bond gets calculated at the earlier nodes. But the following assumptions are in action-optimal play and rationality of players in the future. Moreover, sequential games use it to identify subgame perfect equilibria.

It has implications like analyzing potential outcomes for making informed decisions during uncertain situations and adjustment of investment strategies by investors. It fills and balances information gaps to aid investors in anticipating potential results, leading to optimal portfolio investment options. The financial world employs it in simultaneous moves and sequential games. It also analyses sensitivity and scenario planning to identify optimal pricing strategies and achieve stability.

Backward induction in game theory tries to achieve backward induction equilibrium through dynamic programming, offering a strategic approach to decision-making in sequential scenarios. Additionally, the backward induction practitioners consider different states of the world and their impacts on financial investments, to arrive at the best financial decisions. All these factors contribute to more rational and effective decision-making in the financial world.

### How To Do?

Since it is quite vital in the decision-making process, it is important to know the steps to achieve it effectively:

1. Identify the Final Decision Point: Decide which will be the final choice made in the sequence to start. The process of working backward begins here.
2. Analyze Optimal Choices at the Final Point: At the crucial decision-making stage, decide which options or tactics are ideal for each participant while considering prospective rewards.
3. Work Backward One Step at a Time: Return to the decision point in the previous sequence.
4. Consider Future Choices and Payoffs: Analyze carefully the options and potential outcomes for each choice that might be made at the present decision point in relation to the upcoming decision points that were identified in Step 2.
5. Determine Optimal Choice for the Current Point: Determine which option or course of action is optimal for each player at this particular decision point based on the analysis done in step 4.
6. Repeat Until the Beginning: Steps 4 and 5 should be repeated as you proceed backward through each choice point in the sequence until you arrive at the starting decision point.
7. Optimal Strategy Revealed: The decisions made at every stage of the backward induction procedure will indicate the best course of action for every player for the whole sequence.

### Examples

Let us use a few examples to understand the topic.

#### Example #1

Imagine Ignis and Umbra as competing financial specialists vying for a precious asset in the fictitious financial hub of Lumina. The market in which the competition is held is challenging, risky, and under continual observation. Every action they do may be compared to choosing an investment strategy: they can pick low-risk alternatives to play it safe or take on high-risk actions. Umbra like steady, low-risk surroundings, whereas Ignis thrives in high-risk settings.

In financial markets, the big gain comes from a certain profitable and steady market scenario, but getting there requires some risk-taking. Ignis predicts Umbra's movements using a technique known as backward induction and makes bold decisions. On the other hand, Umbra takes precautions to maintain stability and perhaps prevail. This illustrates the strategic game of backward induction in financial decision-making and highlights the importance of careful preparation and taking measured risks in financial plans.

#### Example #2

The topic discussed here has been examined in an article published on September 12, 2022. Backward induction is defined as using logic to identify the best options at the beginning of an issue from the conclusion. In the 21 Game, players take turns adding 1, 2, or 3 to a beginning value in an attempt to stay under the 21 barriers. Author Eva M. Krockow uses this game as an example to demonstrate this method.

To arrive at a multiple of 4 with every step, players might employ backward induction to ascertain a winning strategy. The article's conclusion emphasizes the need for backward induction to address long-term objectives and make effective decisions.

### Backward Induction vs Forward Induction

Both procedures are recognized for their strategic approach to decision-making. But they have certain differences as follows:

1. What is backward induction in game theory?

In game theory, it is a procedure that entails working backward in time, option by choice, from the conclusion of an issue or circumstance to its inception, to ascertain the best course of action. It examines the optimal course of play for a game by working backward from the last player's move.

2. How to do backward induction in game theory?

Take the following actions:
1. Determine the optimum decisions based on payoffs starting at the game's final decision points.
2. To locate the Nash equilibrium (best strategy for all players), go backward, removing illogical behaviors at each step.
3. Continue until you reach the first decision point.

3. How to solve backward induction?

One must apply the subsequent actions to solve backward induction:
1. Recognize the Game: Identify the Players and their respective Payoffs and Strategies.
2. Start at the End: Choose the best move for each player, starting with the last move.
3. Work Backward: Proceed to the preceding step while taking into account the best move for each participant.
4. Iterate: Continue working backward until you arrive at the first choice, selecting the best course of action at each turn.

This has been a guide to what is Backward Induction. Here, we explain its examples, how to do it, and comparison with forward induction. You can learn more about financing from the following articles –