Simulation Approach for a Two Player Real Options Signaling Game

GEORGIOS RIGOPOULOS

Department of Digital Innovation Management

Royal Holloway University of London

Surrey TW20 0EX

NIKOLAOS V. KARADIMAS

Div. of Mathematics and Engineering Science

Department of Military Sciences

Hellenic Army Academy

Evelpidon Av., 16673

GREECE

Abstract: - Situations where hidden information exists among involved parties can be found in a variety of

diverse domains, ranging, for example, from market entry to military operations. Game theory provides

valuable tools to model and analyze such complex settings, with signaling games being one of the approaches.

Entering an existing market poses several challenges for a new player and can be studied from a variety of

viewpoints. One way to approach it, is by a real options signaling game, where in the simplest form an entrant

and an incumbent firm are participating and hidden information exists. In this paper we focus on the market

entry scenario and approach it by means of a real options signaling game. The work builds on previous work

and contributes to the limited literature on the domain. We introduce the basic notations and background and

describe the game setting. Next, we present a simulation approach demonstrating the basic steps, according to

the theory, and present the results of simulation executions. The work aims to build a generic model for such

market games on top of a two player setting, but the concept is not limited to market entry only, but further

expanded in relevant domains where hidden information exists.

Key-Words: - Real Options, Signaling Game.

Received: June 18, 2021. Revised: February 9, 2022. Accepted: March 20, 2022. Published: April 5, 2022.

1 Introduction

Hidden information is always present in business

environments, where players avoid to disclose

sensitive or private information to their competitors.

Competitors perceive opponents’ actions as signals

of hidden information and they select their

strategies, based on this assumption. Except for the

business, this type of setting is more than obvious at

more critical environments, like military operations,

where privacy of information is secured at any cost.

In such settings, game theory offers a formal

language for analysis and strategy formulation, by

the well-studied class of signaling games.

Signaling games are dynamic games of

incomplete information. The simplest form of a

signaling game comprises of two players who share

different degrees of information. Player one is

considered as the sender, who selects an action,

based on some parameter selected by ‘nature’. This

action is perceived as a signal from the second

player, who then takes an action considering this

signal. At the end, players’ payoffs are calculated

based on their actions and nature’s parameter value.

More general, this type of games can be part of a

strategic setting with players carrying various levels

of information and use their actions as signals to

their opponents to control their access to hidden

information. Research in signaling games is wide

enough and spans to generic problems like labor

markets [1], online auctions [2] and contracting [3],

or more specific areas like real options [4], [5], [6],

[7], [8], [9], [21], [22], [23], [24], [25], [26].

Entry into an existing market poses several

challenges for a new player and can be studied from

a variety of viewpoints. One way to approach it, is

by a real options signaling game, where in the

simplest form an entrant and an incumbent firm are

participating and hidden information exists.

Research in real options signaling games is

relatively limited, despite its potential [4].

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DOI: 10.37394/23207.2022.19.87

Georgios Rigopoulos, Nikolaos V. Karadimas

E-ISSN: 2224-2899

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In this paper we present a simulation model for a

real options signaling game for the marker entry

case. The model and the relevant theory are

introduced, and results are presented from the

simulation. This work builds upon previous research

and contributes in the existing literature by

introducing a relative novel approach ([4], [5], [6],

[7], [8], [9], [10], [11]; [12], [13], [14], [15], [16],

[17], [18], [19]).

2 Background

In game theory, signaling games comprise a special

and very interesting category of games with

incomplete information, which can be applied to a

variety of real world settings. One application

domain is real options, where information

asymmetry exists between players and also there is a

value in real option. For example, a player can apply

a delay option for an investment. The complexity of

this type of games in real options makes hard to

achieve analytical solutions. So, approaches in

literature are relatively limited.

One of the key difficulties to derive analytical

solutions is the stochastic variables that are included

in the model. Infinite future paths can be generated

and, as such, it is not possible to solve the model

and generate equilibrium results. It is possible that

other game parameters may also have asymmetry,

like the players for example, but the most

challenging issue regards the information

asymmetry or imperfect information.

From the literature review, we can see that

almost all works present equilibrium results in the

form of formulas that provide a kind of entrance

threshold for the investment [20]. Very few works

present specific equilibrium and corresponding

payoffs. They present it under certain assumptions

which reduce the infinite values to a finite level.

However, despite the complexity, the domain is

active and recent research works on the field prove

the increasing interest and research importance for

the domain.

In general, this type of games, and subsequent

models, can be distinguished in continuous time

models and models of discrete time. The most

significant contributions are coming from Grenadier

and Watanabe, who work on real options signaling

games in continuous setting, and van de Walle, who

uses discrete time setting [13], [18], [19]. Van de

Walle uses approximation methods and he models a

real option investment game with asymmetric

information as a binomial lattice model. In his

model, the game setting comprises from an

incumbent firm which has the information

advantage, and an entrant firm which lacks

information about a specific parameter, the

investment costs. In this setting, the investment

decision depends on a set of parameters which

include the present value of the project, the binomial

parameters, the market share, the continuous-time

discount rate and of course the private information

which is the investment costs. So, a player, which

represents a firm in the game, will decide to proceed

at an investment if the player expects that the payoff

will be positive. Also, the player expects that his

investment timing is a result of the possible option

value of waiting. We can identify some limitations

in this approach. The key issue is that the game

complexity in the existence of more than three

periods is very high. Also the approach is limited to

two players setting, and the value function is the

present value. However, van de Walle [13] tries to

present an approach, which is mostly applied and

tries to deal with the complexity of real options

signaling games.

This work contributes to existing research, by

introducing a simulation approach for a discrete

time setting in real options signaling games.

3 Formal Game Definition

In its very basic form, a signaling game is a

Bayesian game in extensive form with observable

actions, which comprises of the following:

Player 1, called the “Sender” (S).

Player 2, called the “Receiver” (R).

Random variable , whose support is given by a

set  and is called the type of S (player S knows the

value of t and is considered as private information).

Probability distribution 󰇛󰇜 over T, which

comprises the prior beliefs of player R.

Set  of “Sender” (S) actions (called signals or

messages ).

Set  of “Receiver” (R) actions, with .

Function ℝ (the payoff for

player  at the end of the game).

The timing of the game is as follows: Nature

selects one type  for the Sender (S) from the set

󰇝󰇞 of the feasible types, according to

a probability distribution 󰇛󰇜, where 󰇛󰇜 

and 󰇛󰇜

.

The player with the private information moves

first. The Sender (S) observes  and selects an

action (message) from her set of feasible

actions .

Player 2 with no knowledge of player’s 1 private

information moves second. The Receiver (R)

observes the message  (but does not know

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Georgios Rigopoulos, Nikolaos V. Karadimas

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Sender’s S type:) and selects an action 

from her set of feasible actions 󰇝󰇞.

The game ends and players receive their payoffs,

calculated by the function  and

 accordingly. Player’s 2 payoff

depends on the type of Player1 (ti).

4 Structure of the Market Entrance

Real Options Signaling Game

In order to simulate the investment threshold in a

market, we considered a real options signaling game

where one player represents an incumbent firm and

the other the entrant firm. The following conditions

were set for the game:

A set of players 󰇝󰇞󰇝󰇞 and

Nature, where  is the incumbent firm (Sender),

and  is the entrant firm (Receiver). Nature

selects the type of the incumbent firm.

A random variable t, given by the set 󰇝󰇞

(known to S). It represents the investment cost level

and it can take either the value L (low), or  (high).

Nature selects the actual investment cost at the

beginning of the game and this is known to the

incumbent firm, but not to the entrant. Incumbent

form thus, has private information which is not

known to the entrant firm.

A set of costs 󰇝󰇞 which reflect the

actual investment cost per type t. The investment

cost can take either low () or high value () and

it remains constant during the game. (In case of

investment the two firms face the same cost and

they know the actual values). The values of 

󰇝󰇞 are common knowledge, but the actual

investment cost is known only to the incumbent and

is selected by nature at the beginning of the game.

A probability distribution π(t) over 󰇝󰇞

(the prior probability that the incumbent  is of

type t). The values of π(t) are common knowledge.

A probability distribution for the incumbent

󰇟󰇠 over the set of messages  for every

type (the probability for each message that

the incumbent will send the specific message

conditional on his type . These values are

common knowledge.

A set of entrant’s  posterior beliefs. They

represent entrant’s beliefs about incumbent’s type,

conditional on the message (can be considered as

common knowledge). The entrant firm assigns these

probabilities for every incumbent type and on each

message. Based on this, when the incumbent sends a

message, the entrant firm updates beliefs according

to Bayes rule. These are common knowledge as

well. The beliefs are updated as follows

󰇛󰇜󰇛󰇜

󰇛󰇜󰇛󰇜



where 󰇛󰇜

 .

A set Μ of incumbent’s actions (signals or

messages). They are of type 󰇥󰇦,

where 󰇝󰇞󰇝󰇞󰇝󰇞

(subscripts represent nodes). We allow decisions to

invest , or not invest .

A set  of entrant’s actions (actions). They are of

type 󰇥󰇦, where 

󰇝󰇞󰇝󰇞󰇝󰇞 (subscripts

represent nodes). We allow decisions to invest  or

not invest .

A function R that is the payoff

to player i at the end of the game.

Players’ payoffs are given by the functions

 and  accordingly.

We consider the market as Cournot like and the

demand as 󰇛󰇜 where  is a

stochastic variable (approximated by a binomial tree

evolving over time).

Incumbent’s marginal cost before the investment

is c.

For the payoff functions, the overall approach of

van de Walle [13] was followed.

The solution concept that is used for the game is

the following. We consider an assessment

󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜 which is consisted of the

incumbent’s messages 󰇛󰇜󰇛󰇜 for high and low

type, the entrant’s action 󰇛󰇜 for the incumbent’s

message and the posterior entrant’s belief for the

incumbent’s message and type [4].

5 Simulation Model for the Signaling

Game

In general, an elementary two-period game can be

implemented in a straightforward way using relative

widely used software, such as spreadsheets for both

the analysis and solution [13]. This approach is

sufficient for handling data and low level of

complexity of a two-period game. However, in

order to model a more advanced game, like a multi

period or a multi-player model, some advanced

approach (for example an object-oriented

programming language), should be used in order to

achieve better performance and scalability.

Especially, for the approach we followed using

binomial lattice, the complexity of the game

t

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Georgios Rigopoulos, Nikolaos V. Karadimas

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increases as the periods increase and more players

participate.

For the present work, our approach was to use an

object-oriented language, as it can handle all the

issues and provide a fast and accurate solution. The

model and simulations were developed in C++

language and the graphs were produced by

importing the output data to a spreadsheet program.

Although the model was designed to be generic

enough for quick scaling up to multi periods or

multi players, our initial study was limited to a two-

period, two-player setting. Refinements or

theoretical enhancements will be the subject of

future research. Future work also includes the

presentation of the algorithm in a more formal way

along with metrics of performance and efficiency.

The solution implementation steps are presented

below, following the way they are implemented in

the C++ program. For the two-period game the steps

are the following:

• Step 1: Definition of parameters and

variables.

This is the starting point of the game. In this step

the game parameters and variables of the game are

defined. For the simulation we set the following

initial values to the parameters and derived variables

(Table I):

TABLE I. SIMULATION PARAMETERS

Node 

Parameter/variable

Value

Formula

Value

Risk adjusted discount

rate 

Initial market demand



High cost 

Low cost 

Initial incumbent’s

marginal cost 

Reduced incumbent’s

marginal cost 

Binomial parameters

Risk free rate 

Time 

Periods 

Maturity 





Volatility 

20%





1,492







0,67

Constant asset payout

yield 

 



0,038

Risk neutral

probabilities

Node 

Parameter/variable

Value

Formula

Value



󰇛󰇜



0,401





0,599

Profit share 

0,2

Prior probabilities

󰇛󰇜

0,5

󰇛󰇜

0,5

• Step 2: Assignment of values to parameters

and initiation of variables.

At this step all the parameters are assigned

values according to the model to be studied and the

variables and other structures (such as tables and

arrays) are initialized. All these values comprise

actually the public information that is available to

the players. Prior probabilities are initialized

without a formal procedure. Also, this step can be

executed once, in case of a single play, or recurring

in case of a simulation scenario, where the values

are reassigned partially.

• Step 3: Nature’s selection

This is the step where nature selects the

incumbent’s type. In a realistic scenario the

algorithm uses a random number generator process

to produce the outcome.

• Step 4: Formulation of players’ strategies

per node

This is the step where player’s strategies are

formulated in tables of decisions per node to cover

all possible combinations. This is a dynamic process

and depends on the number of players and nodes of

the game. This is easy to depict in the two-

player/two period game, however in the case of a

multi model complexity increases and depiction is

not always easy.

• Step 5: Entrant’s beliefs update

Here, the entrant’s posterior probabilities are

calculated for all the incumbent’s messages

according to Bayes formula. Although in the game

flow the incumbent selects the message first and the

entrant calculates the posterior belief, for the

solution we calculate the values beforehand.

• Step 6: Entrant’s expected costs

Here, the expected costs are calculated for all the

incumbent’s messages. These are the expected costs

that the entrant assigns to each incumbent’s

message.

• Step 7: Entrant’s expected payoffs per node

Here, the expected payoffs are calculated for all

the incumbent’s messages per node. These are the

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expected payoffs that the entrant calculates for each

incumbent’s message per node.

• Step 7: Entrant’s best expected payoff per

message

Here, the best responses are calculated for all the

incumbent’s messages. These are the maximum

expected payoffs that the entrant calculates for each

incumbent’s message.

• Step 8: Incumbent’s payoffs per node

Here, the payoffs are calculated for all the

entrant’s actions given the incumbent’s messages

per node. These are the payoffs that the incumbent

calculates for each entrant’s action per node.

• Step 9: Incumbent’s best payoff per action

Here, the best responses are calculated for the

entire entrant’s actions. These are the maximum

payoffs that the incumbent calculates for each

entrant’s action.

• Step 10: Entrant’s best payoff

Here, the best entrant’s payoff is calculated from

the best payoffs per action, given the incumbent’s

messages. The action that is selected is the entrant’s

best response.

• Step 11: Incumbent’s best payoff

Here, the best incumbent’s payoff is calculated

from the best payoffs per message, given the

entrant’s best responses. The message that is

selected is the incumbent’s best response.

• Step 12: Solution

The message-action combination is the Bayes

equilibrium with the final payoffs for both

companies.

In case of simulation the above steps are repeated

with modifications of the parameters, which

however do not affect the overall flow and

calculations.

Multi period or multi player setting requires the

repetition of the previous steps, not necessarily in

the same form or sequence, as in this case another

algorithm may be more efficient. However, as said,

it is not within the scope of present work to work on

more complex settings. In addition, the specific flow

has been implemented without any consideration to

performance optimization.

6 Simulation Results for the

Entrance/Investment Threshold

Entrance or investment threshold in a market

definition varies in relevant studies. However, a

common approach is to define it as the demand level

which is appropriate for a firm to enter a market in

order to acquire profits. In the case of continuous

time models the model is solved and it provides the

demand level is a function of a number of model’s

variables and offers a decision rule to the firm as a

threshold to invest. As in continuous time models

equilibrium analysis cannot be provided due to the

infinity of combinations the analytical results are

often limited to the investment thresholds.

In the present study as we follow a discrete

framework, we cannot provide analytical formula

for the threshold however we can simulate the

threshold for various demand values. The threshold

is defined as the demand level for which the payoff

of the firm is larger than zero. As the cost is

included in the calculation of the payoff the

threshold definition is sufficient in order to provide

for an investment decision rule.

So, in a more formal way we can define the

investment threshold as the demand level for which

the payoff is positive (larger than zero). However,

the decision to invest or no, given that the threshold

is reached, depends on the real option value as well.

So, if the payoff is less than the option value

investment can be postponed till the payoff equals

the real option value. In case the payoff is larger

than zero and larger than the option value

investment can be done.

According to this definition we calculate the

investment threshold as the demand value

where the equilibrium payoff of the firms is

larger than zero. In order to identify that value, it is

necessary to calculate the equilibrium payoffs till

the value of payoff turns to a positive value.

So, for the incumbent the demand threshold is

defined as

ℝ

 󰇱

󰇡󰇡󰇛󰇜󰇛󰇜󰇛󰇜󰇢󰇢

 󰇲

As the levels of demand and relevant payoffs are

subject to volatility changes, we run a set of

simulations on the way the investment level changes

when volatility is modified between zero and one

(0%-100%). In the following diagrams we depict the

results of the simulation for various scenarios.

 In Fig. 1, we see the decrease of investment

thresholds for both incumbent and entrant for

volatility increase. The threshold is the

demand value where the above condition is

true.

 In Fig. 2 we see the threshold I for incumbent

in addition to the payoff and real option

value.

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Georgios Rigopoulos, Nikolaos V. Karadimas

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Volume 19, 2022

 In Fig. 3 we see the threshold II for

incumbent in addition to the payoff and real

option value.

 In Fig. 4 we see the threshold for the entrant

in addition to the payoff and real option

value.

Fig. 1. Investment thresholds vs volatility

Fig. 2. Incumbent’s investment threshold I

Fig. 3. Incumbent’s investment threshold II

Fig. 4. Entrant’s investment threshold

From the above results we see that as volatility

increases the thresholds move towards lower

demand values, which is in accordance to the

previous results as well.

The key question of the specific model is when a

firm should invest, or find the optimum strategy,

given the other player’s strategies and beliefs. In the

specific model the payoff is impacted positively by

demand increase, especially for high values of

market demand. It can be argued, thus, that the

higher the payoff the sooner the investment decision

is. However, in the case of competition and strategic

decision making the decision is not based on the

payoff level only and the exact relationship between

payoff and demand depends on the level of the cost

as well it affects the payoff value. In a real-world

scenario, or a richer model, the relationship we

found may not be always the case, or we may

discover fluctuations. Moreover, if we consider both

player’s strategies in the two player/two period

game the relationship between payoff and demand

leads to the presence of three investment zones, the

no-invest/no-invest, the invest/no-invest and the

invest/invest, where the thresholds are variable

according to the values of the rest parameters. In

each zone each firm behaves according to the

equilibrium. Investment decisions thus seem to be

related to investment thresholds which depend on all

game’s parameters values.

Although the present model is not directly

comparable to models presented in literature,

however, partial qualitative comparison may be

done. Grenadier and Watanabe [18], [19], [20]

present continuous time models and they provide

analytical solutions for the threshold values for each

strategy. They do not solve the model but provide

with some numeric examples. The work of van de

Walle [13] is closer to our work and it is a discrete

model with results similar to ours, in terms of

investment strategies and simulation of various

parameters. Zhu [10] on the other hand although

proceeds to analysis without Bayesian update, he

also identifies three regions of equilibrium, which

are close to the zones we identified.

7 Conclusion

In this paper we presented a two period real options

signaling game simulation. The game setting was

introduced along with its parameters and some

results from the simulation were also depicted. From

the model and the simulations, it is evident that the

payoff is impacted positively by increase in demand,

especially when market demand remains high. So,

we can infer that the higher the payoff the earlier the

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Demand value

Volatility

INC Threshhold inc payoff ENT Threshhold ent payoff

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Demand value

Volatility

Threshold 1

INC Threshhold inc payoff ro ro - inc payoff

-5,00

0,00

5,00

10,00

15,00

20,00

25,00

30,00

35,00

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

Demand value

Volatility

Threshold RO

Vola INC Threshhold inc payoff ro ro - inc payoff

-10

-5

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Demand value

Volatility

Threshold ENT

INC Threshhold inc payoff ro ro - inc payoff ENT Threshhold

WSEAS TRANSACTIONS on BUSINESS and ECONOMICS

DOI: 10.37394/23207.2022.19.87

Georgios Rigopoulos, Nikolaos V. Karadimas

E-ISSN: 2224-2899

1005

Volume 19, 2022

investment decision is. However, when we have

competition and strategic decision-making decision

is not based only on the payoff level. The

relationship between payoff and demand depends on

the level of the cost as well it affects the payoff

value. For real world situations, where complexity is

greater, we may meet some fluctuations and

deviations from the above approach. However, the

overall approach seems promising for modeling

some simple scenarios and will be further developed

in future research.

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Georgios Rigopoulos, Nikolaos V. Karadimas

E-ISSN: 2224-2899

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Georgios Rigopoulos carried out the theoretical

analysis and development of the model.

Nikoalos Karadimas carried out simulations and

results analysis.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No sources of funding were used for this artice.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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WSEAS TRANSACTIONS on BUSINESS and ECONOMICS

DOI: 10.37394/23207.2022.19.87

Georgios Rigopoulos, Nikolaos V. Karadimas

E-ISSN: 2224-2899

1007

Volume 19, 2022