Game Theory Analysis

Help for game theory analysis in the real options valuation template for Excel

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The Game Theory model utilizes real option pricing and Nash equilibrium to calculate the expected strategy and payoff for two firms in a competitive environment. The information provided from such an analysis is extremely useful to evaluate proposals to enter a competitive market.  The models works by evaluating each firm's optimal strategy in order to launch the product or service in the market (exercise a call option) in relation to its competitor. Each firm can be either, a leader (enter the market first), a follower (enter after), or both firms could enter simultaneously. The model calculates the probabilities for each of these scenarios and the resulting expected payoff.

Inputs

On clicking the 'Start' button in the 'Menu' sheet; you are taken to the 'Game' sheet.  Inputs are entered into the cells highlighted with a light blue color. Assistance for both input and calculated parameters can be sought by selecting the relevant cell and clicking the Help button to the right. The key inputs required for the model are:

• Investment required. This is the present value of costs required to enter the market with the new product or service. This represents the exercise price on the call option to enter the market. This can be obtained directly from traditional discounted cash flow analysis, such as supplied by the Investment Valuation model.
• Annual Growth Rate of Potential Market Cash Flow. This is the expected grow rate of total market cash flows.  Since this is difficult to estimate, underlying growth of the entire economy can be used as a substitute. Annual Gross Domestic Product (GDP) growth can usually be obtained from National Statistical Offices or third party market data providers.
• Annual Standard Deviation of Cash Flow in Target Market. This represents the volatility of expected cash flows in the market for which the product or service is to be launched. The standard deviation of cash flows can be estimated by one of the following methods, in order of preference:
1. If similar projects or investments have been undertaken or made in the past the standard deviation of cash flows resulting from these projects can be used as a proxy for the standard deviation in cash flows for the proposed investment.
2. Probability analysis can be run on simulations of key inputs, such as revenue and cost drivers, market size and market share, to estimate the standard deviation of the resulting present value. While this type of analysis can be accomplished by using sampling analysis in the Analysis ToolPak add-in shipped with Excel, third-party add-ins can facilitate more sophisticated applications.
3. The standard deviation of publicly traded firms in the same business or industry can be used a proxy for the proposed investment.  This is the least preferred method due to the likely diversity of activities undertaken in other firms and resulting differences in variance characteristics. Such industry specific volatility data can be obtained from third party market data providers (such as those recommended at the Business-Spreadsheets.com web site) and entered into the Pre-Defined sheet for future use across models and proposals. It should be noted that the Pre-Defined sheet can also be utilized to store standard deviation data from similar projects undertaken in the past as described in the first method.
• Inverse Demand Function for 1 firm in market. The inverse demand function specifies how 'elastic' the market is to marketing efforts. The default value of 1 here means that if 1 firm is in the market, it receives 1 times (or 100%) of the market cash flows.
• Inverse Demand Function for 2 firms in market. In a similar fashion to the previous input this specifies the percentage of original market cash flows available to the firm when 2 firms are marketing the product or service. The default value here of 0.75 is based on typical real world scenarios, whereby marketing efforts of two firms increase the market size by 50%. The resulting 150% of the original market size is shared between firms, thereby receiving 0.75 (or 75%) each.
• Risk-free rate. This is the annual risk-free interest rate used in the analysis to create a replicating portfolio consisting of the underlying asset and the risk-free asset. The equivalent 1-year government bond rate can be used here.
• Current expected annual level of cash flow in market. This represents the current annual size of the market for proposed product or service. This can be built up from assumptions using historical data from similar products, market surveys, demographics, or macro economic data. While this is a key input to drive the results of the analysis, sensitivity analysis is undertaken to assist in understanding its impact.

Parameters are calculated from these inputs to drive the analysis. These are:

• Level of annual market cash flow for Follower to enter market. This is the total annual market cash flows required for the following firm to enter the market (i.e. after another firm has already entered).
• Follower's Payoff Value. This is the expected present value of cash flows available to the following firm.
• Value of Leader's future profit flow. This represents the present value of cash flows that the leading firm will receive assuming the following firm undertakes a following strategy. The Leader receives monopoly cash flows from the market until the market size reaches the level where it makes sense for the Follower to enter. After that, both firms receive duopoly cash flows.
• Leader's Payoff Value.  This represents the value of the Leaders future profit flow less the investment required to enter the market (exercise price).
• Value if both firms exercise immediately and simultaneously. This considers the present value of cash flows available to each firm if they both enter the market with the product or service at the same time.
• Simultaneous Payoff Value. This represents the value of the cash flows entering the market simultaneously with the competitor less the investment required to enter the market (exercise price).
• Annual market cash flow where value of Leading equals Following. This represents the unique annual level of market cash flow where the value of being a Leader equals the value of being a Follower. In order to calculate this parameter (under the Nash equilibrium), the Solve program needs to be run by clicking the Solve button to the left. If any of the inputs are subsequently changed, 'Required' will appear in red text underneath this button to indicate that the Solve program needs to be rerun.

Sensitivity Table

Below the main parameters section is a table displaying the strategy probabilities and resulting expected payoff for different levels of annual cash flow in the market.  The first row shows the expected level of market cash flows and corresponding results. Next, two rows show levels of annual market cash flow is below the level where leading equals following. In this case, each firm will wait until cash flow reaches this level, which is displayed underneath.

Below this two more levels of annual cash flow are shown greater than the level where leading equals following and less than the level where it pays to follow.  In this region, each firm will consider leading and the probability to exercising increases as the level of cash flows available increases.

Finally, the level of annual market cash flows where it pays to be a follower is displayed with two levels greater than this underneath. For these levels, both firms with enter the market immediately and simultaneously.

Results Summary

Below the sensitivity analysis is a textual summary of results for the expected annual level of market cash flows. This provides an explanation of each of the key parameters, strategy probabilities for each firm, and resulting expected payoff. This summary can be useful for direct insertion into business case proposals and reports.

Screenshot: Game Theory Analysis

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