Game Theory and Family Business Succession

Repeated and Extensive Games

Repeated and extensive games are particularly well suited to issues in family business succession. The structure of repeated games is no different than that of normal games. The primary difference between them resides in the number of times the actors play the game. Normal games are typically thought of in terms of single-shot games: the actors are assumed to play the game once. Repeated games, alternatively, consider situations in which the actors play the same game multiple times. Actors have the opportunity to learn in repeated games.

Importantly, repeated games provide a methodology for managing the difficulties arising from variance and moderating factors, both noted above, in estimating actors’ payoffs and utility functions. In essence, both variance and moderators challenge assumptions of perfect knowledge and rationality in game theory. These challenges are especially important if the actors in an interaction only engage with one another one time. However, a repeated game provides opportunities for the actors in a game to learn about the choices that the other actors are likely to make.

In this respect, repeated games allow for interventions. That is, interactions modeled initially as leading to conflict or indecision may be changed in some way over time as to avoid those difficulties. Game theory analysis has the potential to identify the underlying causes of difficulties in succession events and possible paths to avert them.

Extensive games are those in which actors move in succession, allowing for consideration of various scenarios through what-if and if-then statements. Unlike strategic and repeated games, extensive games allow for variations in the conditions in which an actor’s decisions are made. As the game progresses through levels of actors’ decisions, the basis for future decisions changes. Extensive games are often modeled as decision trees.

An example will help explain extensive games. A simple extensive game might describe a situation in which the CEO of a family business has a daughter and a son, and values education. The daughter, at 27 years old, has shown the most interest in the firm but has only made partial progress toward college degree even though she is capable of doing the work; the son has shown less interest in the firm but has graduated from a good university and would join the firm if his father pressed him. The CEO would be happy to pass the firm to his daughter and would be most pleased to give the job to a person who really wants it. However, he feels responsible for the future of the firm and will ask his son to take the position, even if he does not prefer it, to promote the firm’s success. See Figure 2 for a graphic on this game.

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Figure 2.

Extensive game.

In this case, the CEO’s payoffs might be structured as dependent on the successors’ actions through a game tree model of an extensive game. The payoffs are presented as the numerical outcomes for the daughter and CEO, respectively, associated with each path. An additional game tree could be developed for the interaction between the CEO and his son.

Notice that the founder’s basis for decisions changes as time passes and the children make their own decisions. In some respects, the founder has greater information on which to base decisions; on knowing whether or not the daughter is successfully completing her education, his future decisions will be more certain. The degree to which clear paths arise from the game trees indicates the degree to which such situations would involve conflict or indecision.

Problem Statement

Management succession events result in the immediate or eventual transfer of management control of a family business. Please note that since we are concentrating on management succession, we are assuming the family, including the CEO, is committed to retaining ownership in the firm regardless of who is managing it. Relaxing this assumption is not problematic for game theory, but the complexities of doing so would distract us from the real purpose of this article: creating a foundation for the use of game theory for understanding management succession in family business.

Actors

Although family businesses are also composed of other important parties, such as spouses, nonsuccessor siblings and relatives, nonfamily managers, and a host of other stakeholders, the games described here are restricted just to CEOs and one or more potential successors. The influence of these other parties will be considered when constructing each actor’s strategies, payoffs, and utility functions.

Conditions

The type of game being played (normal, repeated, or extensive) and the strategies available to each actor must be stipulated. To analyze conflict and indecision in a planned succession event on a cross-sectional basis, normal games likely will be the most appropriate structure. Analysis of succession events over time under the assumption that the characteristics of the actors do not change materially but that they do learn from one another would likely be structured as a repeated game. Successions that involve important changes over time, such as an increase in the education or experience of one of the potential successors, will likely be structured as extensive games.

Payoffs

As noted above, payoffs describe the outcomes of the game to an actor in the event of a particular set of interactions and are captured in the form of a utility function. Each actor’s utility function represents the series of values associated with each possible strategy available to that actor. There are at least three dynamics that are important in constructing an actor’s utility function.

The first, and most obvious, involves capturing the factors that will influence each actor’s utility. There are many elements that might influence the views of an actor, and the importance of those factors is likely to vary with each actor. The succession literature suggests that the following characteristics are likely to be important, although this list is not exhaustive:

Capturing all of these elements requires some broad thinking. Computing payoffs requires delving into human characteristics such as desires, emotions, and personal feelings that are very hard to quantify. Rather than considering a single preference score associated with one of an actor’s available strategies, it is better to think of a score as the summary of a vector of scores across the factors listed above. Game theory is robust even when preference vectors are complicated. Although enumerating the true payoffs of an actor in a succession event is unlikely, estimations of emotional preferences are evident in almost all management studies. Game theorists have addressed similar problems through the use of “fuzzy numbers,” or a small set of numbers that are related to a true value (Buckley, 1985). This actually helps assure that the game captures the nuances involved when people make important decisions.

Consider a simpler situation involving a major decision, such as shopping for a car. At the beginning of the process, the buyers face a dizzying array of options. They often narrow their choices down to a few models and colors through rational analysis and pragmatic factors, and then make a decision. When asked why they selected one car over another on the short list, the responses will typically have less to do with attributes then feelings. In the end, the purchasers internalized the myriad factors that could be important to buying a car and came up with a single preference. This is an analogous (albeit more complex) situation.

Second, there may be elements that act as moderators of the relationship between an actor’s utility function and his/her actual choice of strategies. A moderating factor is any element that affects the relationship or interaction among any two or more variables. In this case, a moderator would take the form of some sort of influence that changes the relationship between an actor’s maximization of his or her utility function and the choices that he or she makes in the succession event. Moderators may arise both from other actors within the game as well as from forces external to the game. For instance, an actor’s payoffs may be altered by his or her beliefs about the preferences of other actors in the game. For instance, consider a CEO who inwardly believes that a younger daughter would be the best choice to run the firm but publicly supports an elder son who is highly interested in taking over the firm.

Last, people who are not directly involved in the game may influence the game’s actors. That is, people who are important to actors may alter the true utility functions of an actor. For instance, a spouse or other family members may influence a CEO’s decisions. Important customers may force a CEO to alter the selection of a successor, or a bloc of strong nonfamily managers may persuade a CEO to select a member of their group to lead the firm even if the CEO’s preference is to pass the firm to an offspring. A CEO approaching or even past retirement age may acknowledge the need to appoint a successor, but may be internally reluctant to leaving the business.

Interestingly, the degree to which actors face troubles clearly identifying their payoffs is indicative of troubles in the succession process. Many problems in succession arise when actors really do not know what they want, or change their opinions over time. We expect that (a) the less certain or the more equivocal an actor’s payoff vector, or (b) the more an actor changes preferences (in repeated and extensive games), (i) the greater the degree of indecision that will exist throughout a succession process and (ii) the less successful a succession event will be.

Decisions

Analysis of the strategic form, which integrates the utility functions of the game’s actors in the form of a matrix or game tree, leads to indications of what the actors will do in the game.

At this point, game theory analysis turns to the search of dominant strategies and points of equilibrium. A dominant strategy, as noted above, always provides an actor with superior outcomes than any other available strategy. A Nash equilibrium exists at a position in a game in which no single actor would chose an alternative strategy, given the strategy choice of the other actors in the game.

Harmonious succession processes and events are easy to recognize in a game theory analysis. When each actor has a dominant strategy, and those dominant strategies match up to create an equilibrium outcome, there will be no doubt as to the right thing to do. For instance, if a CEO always prefers to pass the firm to his oldest child, and all of the CEO’s other children agree that the eldest is the best choice of a successor, everything will go smoothly.

Harmonious successions, we expect, are the exception rather than the norm. Things are rarely so easy. Consider a few possible scenarios that made result in indecision and conflict among family members (the actors):

Each of these situations shares commonalities. Because these games are likely to end with an option that is not optimal for one of the players (perhaps all of them), conflict and indecision are likely to result. Conflict among actors will come up because succession processes and events involve highly personal, emotional, and professional decisions. They are so important to some actors as to have lifelong impacts on their relationships with other family members and even the business itself. If things do not go as an actor wants or expects, internal feelings of disappointment and external reactions of conflict should be expected. Faced with the potential for conflict, and knowledge that is likely to arise if certain decisions are made, actors may be indecisive in selecting strategies. That is, it is perfectly reasonable for a CEO (father/mother) or a sibling to feel empathy when they prefer an option that they know is not in alignment with the wishes of other family members, and to feel conflicted about those situations. Indecisiveness is sure to follow.

In more formal terms, we expect that the degree of conflict among actors will be lowest when there is only a single Nash equilibrium in the strategic form and significantly increase when there are no Nash equilibrium or multiple Nash equilibriums.

Total Family/Firm Welfare

Game theory typically focuses on the outcomes of interactions among players from the perspective of each individual actor: the results are framed in terms of the payoffs to each individual. In the case of family business succession, we must also consider a more holistic view of the payoffs. That is, we are interested not just in maximizing the payoffs of any given actor but also in maximizing the outcomes for the firm. Successful successions result in a strong future for the family business, regardless of whether a particular player is happy with the outcome or not.

Game Structure

Consider a situation in which a founder of a business (denoted F) desires to pass control of a family business over to one of his two offspring, either his son (denoted S) or his daughter (denoted D). Recall that we are concentrating on management succession, for purposes of simplicity, eliminating from our analysis the option of selling the firm.

The two candidates for succession (i = {S, D}) are each characterized by a level of talent for running the business (denoted t _i ≥ 0) and a level of desire for runn ing the business (denoted d _i ≥ 0). The founder places a positive value on both the level of talent and level of desire of the chosen successor.

Choosing to pursue the CEO position at the family business constitutes an opportunity cost to that candidate. Someone talented enough to be considered for the top position at a family business is likely to be capable enough to get a very good job at some other firm. Suppose that S and D simultaneously choose to either “pursue” or “abandon” the position of CEO. After these choices are made, F then chooses a successor from among those offspring that pursued the position.

If both S and D chose “pursue,” then F could choose either one to be the successor. If one chose “pursue” while the other chose “abandon,” then the one who chose “pursue” will be named the successor. If neither chose “pursue,” then F will appoint an outside (i.e., nonfamily member) successor. These latter assumptions on the choice by F can be justified by supposing that this “outside option” is always available, but the preference for appointing one of the offspring as the successor is so strong that the outside option will only ever be used in a situation in which neither offspring chooses to pursue the position.

This situation naturally can be modeled as a three-player game. The three players are S, D, and F. The strategies available to the players and the timing of the decisions have already been alluded to above. First, S and D each simultaneously choose to either “pursue” or “abandon.” Second, after observing these choices, F then chooses a successor from among those individuals who chose to pursue the position.

Each of these three players has a payoff function. The payoff function for F can be written as π _F = Ld _i + Vt _i from appointing an offspring with desire of d _i and talent of t _i as the successor. The parameter L ≥ 0 captures the degree to which F values naming a successor with a greater desire for running the business (i.e., think of this parameter as capturing the degree to which the founder views the family business as a legacy to be passed on to the individual who really wants to run the business).The parameter V ≥ 0 captures the degree to which F values naming a competent successor who can fully realize the firm’s performance. For instance, a founder who cares primarily about the legacy value of passing control to an interested successor and only slightly about maximizing the future value of the firm would have a relatively large value of L and a relatively small value of V. Finally, suppose that naming an outside successor would give F a payoff of π _F = 0, which follows with our assertion that a family CEO will always prefer to pass the business to an offspring if there is an option to do so. Turning attention to the payoff of candidate i (i = {S, D}), suppose that the value the individual places on being the successor is directly measured by d _i . Further suppose that the individual must incur a cost of c > 0 in order to “pursue” the position. That is, choosing “pursue” and being named the successor results in a payoff of π _i = d _i - c; choosing “pursue” but not being named the successor results in a payoff of π _i = −c; and choosing “abandon” results in a payoff of π _i = 0. We will focus on situations in which each candidate has d _i > c, so that choosing “pursue” may possibly be best for the candidate. This assumption eliminates the uncomfortable situations in which a successor pursues the top job only because s/he has nothing better to do.

Perfect Information

Begin by considering a situation of perfect information, in which all three players know the exact values of d _i and t _i for both i = {S, D} and all of the values of all of the parameters in the payoff functions of each of the three players. ¹ Interestingly, even in this somewhat basic situation, qualitatively different equilibria can result.

To identify equilibrium strategies for this game, we use simple backward induction. That is, optimal behavior must first be determined starting in the terminal node of the game (i.e., the point at which F must choose a successor from among those offspring that expressed an interest in the position). If neither the son or daughter expressed an interest in the position, then F’s only choice is to name an outside successor and realize a payoff of π _F = 0. If offspring i chose “pursue” while offspring j chose “abandon,” then F will appoint i and realize a payoff of π _F = Ld _i + Vt _i . Finally, if both S and D chose “pursue,” then F will choose to name the successor that makes his own realized payoff larger, based on the actual levels of talent and desire of the two candidates, along with the relative value that he places on these two attributes. In terms of the equations specified thus far, F would name D as the successor if and only if ²

L d D + V t D ≥ L d S + V t S ⇔ L ( d D − d S ) ≥ V ( t S − t D ) .

From this inequality, the choice of F is rather trivial if either (a) t _D ≥ t _S and d _D ≥ d _S or (b) t _S ≥ t _D and d _S ≥ d _D . In the former case, F would clearly choose to appoint D over S, since the daughter is both more talented than and has a greater desire than the son. Similarly, in the latter case, F would clearly choose to appoint S over D, since the son is both more talented than and has a greater desire than the daughter. This is the definition of a harmonious succession.

The more interesting situations are those in which either (c) t _D ≥ t _S but d _S ≥ d _D or (d) t _S ≥ t _D and d _D ≥ d _S . In these cases, the choice by F depends on the relative degree to which he values “desire” vis-à-vis “talent” of the successor (i.e., the values of the parameters L and V) and the differences in the levels of these attributes between the two candidates, that is, (t _D −t _S ) and (d _S −d _D ).

However, with perfect information not only can F easily make this comparison and choose the preferred successor if both candidates choose “pursue,” but both D and S can make the comparison in the inequality above and infer who will be named the successor if both choose “pursue.” Thus, in the initial stage during which D and S simultaneously choose to either “pursue” or “abandon,” one of two simultaneous move games is relevant.

If L(d _D - d _S ) ≥ V (t _S −t _D ), then D and S are essentially playing the simultaneous move game depicted in Figure 3. The payoffs in this matrix reflect the fact that if both candidates choose “pursue” when L(d _D −d _S ) ≥ V (t _S −t _D ), F would name D as the successor. To fully understand this inequality, consider a case in which d _D ≥ d _S and t _S ≥ t _D (i.e., the daughter has greater “desire” while the son has greater “talent”). In light of these differences in attribute levels, the founder will prefer to appoint the daughter as the successor, so long as he cares sufficiently enough about the legacy value of passing control to an interested successor and only slightly about maximizing the future value of the firm (recall, a founder with such preferences has a relatively large value of L and a relatively small value of V). The inequality L(d _D −d _S ) ≥ V (t _S −t _D ) simply provides a precise condition (in terms of the parameters of the founder’s payoff function and candidate attribute levels) for when this is the case.

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Figure 3.

Simultaneous move game: Founder values desire more than competency.

Focusing on the payoff matrix in Figure 3, recall, we are assuming that d _D > c > 0 and d _S > c > 0, or that the successors both have costs of pursuing the job that are greater than zero and their desire to take the job exceeds these costs. Thus, Figure 3 reveals that S does not have a dominant strategy (if D chooses “pursue,” then the best reply for S is “abandon,” whereas if D chooses “abandon,” then the best reply for S is “pursue”). However, for D, “pursue” is a dominant strategy: it is the best reply for D for each of the two strategies available to S. As such, this game has a unique Pure Strategy Nash Equilibrium, characterized by D choosing “pursue” and S choosing “abandon” followed by F selecting D to be the successor.

If, instead, L(d _D −d _S ) < V (t _S −t _D ), then D and S are essentially playing the simultaneous move game depicted in Figure 4. The payoffs in this matrix reflect the fact that if both candidates choose “pursue” when L(d _D - d _S ) < V (t _S −t _D ), F would name S as the successor. To gain insight into this condition, again consider a case in which d _D ≥ d _S and t _S ≥ t _D (i.e., the daughter has greater “desire” while the son has greater “talent”). Faced with such differences in attributes, the founder will prefer to appoint the son as the successor, so long as he cares sufficiently enough about maximizing the future value of the firm and only slightly about the legacy value of succession (recognize, a founder with such preferences would have a relatively large value of V and a relatively small value of L). The inequality L(d _D - d _S ) < V (t _S −t _D ) simply provides a precise condition (again, in terms of the parameters of the founder’s payoff function and candidate attribute levels) for when this is true.

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Figure 4.

Simultaneous move game: Founder values competency more than desire.

When L(d _D −d _S ) < V (t _S −t _D ), D does not have a dominant strategy: if S chooses “pursue,” then the best reply for D is “abandon,” whereas if S chooses “abandon,” then the best reply for D is “pursue.” However, for S, “pursue” is a dominant strategy, as “pursue” is the best reply for S for each of the two strategies available to D. As such, this game has a unique Pure Strategy Nash Equilibrium, characterized by S choosing “pursue” and D choosing “abandon” (followed by F selecting S to be the successor). That is, the game depicted in Figure 4 and the resulting equilibrium is essentially the mirror image of that depicted in Figure 3.

At this point, we can begin to see the powerful insights that can be obtained by formally modeling this situation with the tools of game theory. Under perfect information, the equilibrium is such that one and only one of the offspring pursue the position and the chosen successor is the individual which makes the payoff of F as large as possible. Game theory allows each decision maker to precisely see how the situation will unfold. In this simple example, this ultimately allows S and D to essentially coordinate their actions (i.e., one chooses “pursue,” while the other chooses “abandon”), resulting in a more harmonious outcome.

Furthermore, by parameterizing the payoffs of each player and formally solving for the unique equilibrium, we can gain insight into how the outcome of the succession process would potentially change if the priorities of the various players were to take on different values. For example, again focus on a situation in which dD ≥ d _S and t _S ≥ t _D , and recall that in equilibrium the daughter will be chosen as the successor if and only if L(d _D −d _S ) ≥ V (t _S −t _D ). From here we can directly see how the realized outcome depends critically on the relative degree to which the founder values talent vis-à-vis desire. If, all other factors fixed, the founder values talent to a greater degree (i.e., has a larger value of V), then this condition is less likely to hold, implying that the son (in this case, the more talented candidate) is more likely to ultimately be appointed. If instead, all other factors fixed, the founder values desire to a greater degree (i.e., has a larger value of L), then this condition is more likely to hold, implying that the daughter (in this case, the candidate who wants the position more) is more likely to ultimately be appointed.

Similarly, we can easily see how varying the attributes of the candidates would impact the equilibrium outcome. ³ If, all other factors fixed, the degree to which the daughter desires the position were increased (i.e., if d _D were larger), then the condition above is more likely to hold, implying that she is more likely to be named the successor. Similarly, if, all other factors fixed, the degree to which the son desires the position were increased (i.e., if d _S were larger), then the condition above is less likely to hold, implying that he is more likely to be named the successor.

If, all other factors fixed, the talent of the daughter were increased (i.e., if t _D were larger), then the condition above is more likely to hold, implying that she is more likely to be named the successor. Similarly, if, all other factors fixed, the talent of the son were increased (i.e., if t _S were larger), then the condition above is less likely to hold, implying that he is more likely to be named the successor.

These insights on how the equilibrium depends on talent levels are of tremendous importance for at least two reasons. First, individuals often have actions that they could take (e.g., pursuing an MBA), which would directly increase their talent for managing the business. Thus, it would seem reasonable to suspect that, at least to a degree, a portion of talent is under the control of the individual. ⁴ Second, in terms of ultimately using a theoretical model as the basis for an empirical study, proxies for talent (e.g., level of education or within-industry work experience) can be measured and observed.

Finally, recognize that even though the unique equilibrium results in the appointment of the successor that makes the payoff of F as large as possible, this outcome may not maximize “Total Family/Firm Welfare” (i.e., the sum of payoffs across all players in the game). To see this, consider a situation in which t _D < t _S and d _D > d _S . With perfect information, a prerequisite for maximizing Total Family/Firm Welfare is that only one of the two offspring should pursue the position (since only one individual will be named successor, having both choose “pursue” would simply result in additional costs of c, relative to an outcome in which the individual not named successor instead chooses “abandon”). With this in mind, the efficient outcome can be determined by comparing Total Family/Firm Welfare from realizing “D pursue, S abandon, D chosen” to Total Family/Firm Welfare from realizing “D abandon, S pursue, S chosen.” For the former outcome, F realizes a payoff of Ld _D + Vt _D , D is named the successor and realizes a payoff of d _D –c, and S chooses “abandon” and realizes a payoff of 0. This results in Total Family/Firm Welfare of

( L d D + V t D ) + ( d D − c ) + ( 0 ) = ( 1 + L ) d D + V t D − c .

For the latter outcome, F realizes a payoff of Ld _S + Vt _S , D chooses “abandon” and realizes a payoff of 0, and S is named the successor and realizes a payoff of d _S – c. This results in Total Family/Firm Welfare of

( L d S + V t S ) + ( 0 ) + ( d S − c ) = ( 1 + L ) d S + V t S − c .

Comparing these two expressions, Total Family/Firm Welfare is maximized by “D pursue, S abandon, D chosen” if and only if

( 1 + L ) ( d D − d S ) ≥ V ( t S − t D ) .

Recall, the equilibrium outcome is “D pursue, S abandon, D chosen” so long as

L ( d D − d S ) ≥ V ( t S − t D ) .

For d _D > d _S it follows that (1 + L)(d _D −d _S ) > L(d _D −d _S ). Thus, three outcomes are possible. If V (t _S −t _D ) ≤ L(d _D -d _S ), then the equilibrium outcome of “D pursue, S abandon, D chosen” maximizes Total Family/Firm Welfare. If instead V (t _S −t _D ) > (1 + L)(d _D −d _S ), then the equilibrium outcome of “D abandon, S pursue, S chosen” maximizes Total Family/Firm Welfare. However, if

L ( d D − d S ) < V ( t S − t D ) ≤ ( 1 + L ) ( d D − d S ) ,

then the equilibrium outcome is “D abandon, S pursue, S chosen,” while the outcome that would maximize Total Family/Firm Welfare is “D pursue, S abandon, D chosen.” When the equilibrium outcome is inefficient, feelings of indecision and conflict are more likely to come out of the succession process.

Recognize that these conditions for inefficiency can be expressed as

0 < V ( t S − t D ) − L ( d D − d S ) ≤ ( d D − d S ) .

The term V (t _S −t _D ) −L(d _D −d _S ) is simply the net gain for F from having S be the successor instead of D. The first inequality in the condition above (i.e., the condition that this term is positive) is simply a direct reflection of the fact that F would choose S as the successor over D in the situations being considered. This comparison is based solely on the different possible realizations of F’s payoff.

However, Total Family/Firm Welfare depends not only on the payoff of F but also on the realized payoffs of S and D. For “D pursue, S abandon, D chosen” the sum of the payoffs of the two candidates is d _D −c, while for “D abandon, S pursue, S chosen” the sum of the payoffs of the two candidates is d _S −c. Thus, the net gain in welfare for the candidates collectively from realizing “D pursue, S abandon, D chosen” instead of “D abandon, S pursue, S chosen” is d _D −d _S , which is the final term in the conditions above.

As such, the conditions identifying when the equilibrium will be inefficient intuitively reveal why the equilibrium outcome fails to maximize Total Family/Firm Welfare. When V (t _S −t _D ) −L(d _D −d _S ) > 0, the founder maximizes his own payoff by choosing S over D and the equilibrium outcome is “D abandon, S pursue, S chosen.” But Total Family/Firm Welfare is instead maximized by “D pursue, S abandon, D chosen” precisely when the net increase in the joint welfare of the candidates from realizing this latter outcome (i.e., d _D −d _S ) exceeds the net gain for the founder from realizing the former outcome, that is, V (t _S −t _D ) −L(d _D −d _S ).

Imperfect Information

In this section, we consider a generalization of the game discussed above by considering the impact of imperfect information. By extending the model in this direction, we will ultimately see how altering a basic assumption can drastically change the qualitative nature of the equilibrium. More precisely, in this more general framework the game no longer has a unique equilibrium in which the son and daughter are able to essentially coordinate their actions. This discussion directly reveals the value of formally modeling and analyzing the situation with the tools of game theory, since the important role that information can potentially play is only seen by such an analysis.

Consider a situation in which there is imperfect information, in that none of the players know the true values of t _S and t _D at the start of the game. The values of d _S and d _D are still observed by all players at the start of the game, as are the values of L, V, and c. However, the “true value” of t _i is only revealed if candidate i chooses “pursue.”

For instance, suppose D and S each work at the family business as young adults. During this time they each come to learn, and reveal to others, their desire to run the family business in the future. However, these young adults will not be able to fully demonstrate their ability to be the next CEO. Later, likely when the founder initiates a succession process and the candidates must firmly decide whether or not to “pursue” the top job, each candidate may then have opportunities to take on a more serious role in the business, including more significant and substantial duties. As a result of having to perform these more important functions they learn, and it is revealed to others, their potential for actually running the business.

If neither offspring chooses “pursue” or if only one offspring chooses “pursue,” then the choice of D at the terminal node of the game is still trivial. Thus, for each candidate i, the best reply to a choice of “abandon” by the other candidate (−i) is a choice of “pursue.”

If both offspring choose “pursue,” then the true values of t _S and t _D are revealed, and F can easily choose which candidate to appoint as CEO (naming D the successor if and only if L(d _D −d _S ) ≥ V (t _S −t _D ), which is the same condition that was previously derived). But, when faced with the choice of “pursue” or “abandon,” each candidate i is unaware of both their own talents and those of the other sibling (denoted t _i and t_−i). This is a common occurrence, as the time, responsibility, and dedication required of a family business CEO are significant, and may raise feelings of doubt or insecurity within the candidates.

As a result, when considering the outcome when both candidates choose “pursue,” each candidate can at best determine a value for his expected payoff incorporating the probability with which he expects to be chosen as the successor if both he and his sibling choose “pursue.” To formulate such a probability and expected payoff, a candidate must have some beliefs over the distribution from which the values of t _i and t_−i are determined. To simplify the analysis, suppose that each t _i is determined as an independent realization of a random variable distributed according to the cumulative distribution function F(t) = t (i.e., each talent level is an independent draw from a U[0,1] distribution).

The condition specifying when D will be chosen as the successor (if both D and S choose “pursue”) can be expressed as t S ≤ t D + L V ( d D − d S ) . Without loss of generality, suppose d _D ≥ d _S . The choice by F of who to select as CEO when both choose “pursue” as a function of the realized values of t _D and t _S is illustrated in Figure 5. Under the assumption that t _D and t _S are independent draws from a U[0,1] distribution, each point in the unit square is equally likely. Thus, from the perspective of the offspring when making their initial choices of “pursue” or “abandon,” the probability that D will be selected is

P ( D ) = 1 − 1 2 [ 1 − L V ( d D − d S ) ] 2

and the probability that S will be selected is

P ( S ) = 1 2 [ 1 − L V ( d D − d S ) ] 2 .

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Figure 5.

Founder decision structure under conditions of imperfect information.

From here, it follows that if both choose “pursue,” then the expected payoff of D is

π D p , p = d D ( 1 − 1 2 [ 1 − L V ( d D − d S ) ] 2 ) − c

and the expected payoff of S is

π S p , p = d S ( 1 2 [ 1 − L V ( d D − d S ) ] 2 ) − c .

To ease the discussion, restrict attention to a situation of imperfect information in which D and S have equal desires to be named as the successor: d _D = d _S = d > c. ⁵ In such a situation, if both “pursue,” then the offspring with the larger realized value of t _i will be named the successor, so that P ( D ) = 1 2 , P ( S ) = 1 2 , π D p , p = 1 2 d − c and π S p , p = 1 2 d − c It follows that when making the initial decisions of “pursue” versus “abandon,” it is now as if the offspring are playing the simultaneous move game illustrated in Figure 6. For this game, it is still true (by the assumption of d > c) that for each player choosing “pursue” is a best reply to a choice of “abandon” by the other player. Thus, there are essentially two qualitatively different cases of interest.

OPEN IN VIEWER

Figure 6.

Game under conditions of imperfect information.

First, consider 1 2 d − c > 0 (or equivalently d > 2c). In this case, the desire of each offspring to run the business is sufficiently high so that choosing “pursue” is a dominant strategy for each individual. The unique Nash Equilibrium is characterized by both D and S choosing this dominant strategy of “pursue.” After these choices, the true values of both t _D and t _S are revealed, and the individual with the larger value of t _i (i.e., the higher level of talent) is chosen to be the successor. This outcome is desirable in that we never realize an ex post inefficiency similar to that described above. This is because for d _D = d _S = d, either D or S would realize the same benefit from being named the successor. So, total welfare from the succession decision are maximized by always appointing the individual that is strictly preferred by F, who after all is the one making the decision.

However, at this equilibrium we always have both offspring incurring the costs of c > 0 associated with pursuing the position. If it turns out that t _D and t _S are similar in value (so that |t _D -t _S | is relatively small), then the gain from appointing the offspring of higher talent might not necessarily outweigh the additional expense of having both individuals pursue the position.

Finally, note that in this case there will always be an offspring who actively pursued the position but was ultimately not chosen. This can be thought of as a situation in which there is a “Frozen-Out Successor,” in that the founder has identified one of the two siblings as the choice to be the next CEO of the firm, and the chosen sibling is interested in the job but the other sibling disagrees with the selection and also wants the job. Subsequent conflicts would likely arise between the rejected sibling and both the founder and the chosen sibling.

Second, consider 1 2 d − c < 0 (or equivalently d < 2c). In this case, the desire by each offspring to run the business is sufficiently low so that choosing “pursue” is not a best reply to a choice of “pursue” by his sibling. That is, the value of being the chosen successor is not large enough to warrant incurring a cost of c > 0 with certainty if there is only a probability of ultimately being named the successor. Neither player has a dominant strategy. The game has two Nash Equilibria in Pure Strategies (one in which D chooses “pursue” and S chooses “abandon,” and a second in which S chooses “pursue” and D chooses “abandon”), along with a third Nash Equilibrium in Mixed Strategies. This Mixed Strategy Nash Equilibrium involves each player randomizing between the available pure strategies, with each offspring choosing “pursue” with probability 2(d−c)/d and “abandon” with (2c −d)/d. ⁶

Focusing first on the two Pure Strategy Equilibria, in these situations only one of the two offspring chooses to pursue the position. This does have the advantage of reducing the total costs of pursuit (since the costs of c > 0 are only ever incurred by one of the two individuals). However, the probability that the individual ultimately appointed to be the successor is the one with the higher actual, true value of t _i is only . That is, in this case, the ideal candidate (i.e., the one whose appointment would maximize the payoff of F, and since d _D = d _S = d, would also maximize the sum of benefits over all players) is only appointed half of the time.

Switching attention to the Mixed Strategy Equilibrium, observe that under this pair of strategies, each of the four possible outcomes (of “D pursues while S abandons,” “S pursues while D abandons,” “both pursue,” and “both abandon”) is realized with positive probability. If both “pursue” or if one chooses “pursue” while the other chooses “abandon,” the outcome is qualitatively similar to the respective discussions above. However, for this Mixed Strategy Equilibrium an outcome in which both D and S choose “abandon” also occurs with positive probability. This can be viewed as a situation in which (based on the chosen actions of the offspring) there are “Disinterested Successors,” in that the founder prefers to pass leadership to one of his offspring, but no such individual has expressed an interest and taken the necessary actions to pursue the position. This is clearly an undesirable outcome from the perspective of the founder.

As illustrated by this analysis, with imperfect information, the equilibria of the game (with either d-c > 0 or d−c < 0 ) differ drastically from what arose in the setting of perfect information. ⁷ This result again illustrates the benefit of using game theory to examine this decision making environment. The important role that a factor such as differences in information can play in influencing the outcomes of the family business succession process can only be clearly seen by modeling and analyzing the situation through the lens of game theory.