(Interlude) Strategic Choices in a Paradoxical World: Insights from Arrow's Theorem
How Arrow’s Impossibility Theorem, a famous result in social choice theory, applies to strategic decision-making and why it is important to take decisive action with respect to choices.
In this interlude between editions, I pause to reflect on a fascinating result in social choice theory: Arrow’s Impossibility Theorem. This theorem has been widely adopted by economists and political theorists, yet it remains largely overlooked by policy makers and strategists. In the face of challenges, strategizing about choices requires decisive action, for there is no time to waste.
What does Arrow’s Theorem tell us about how we rank and pursue choices? How can we, as a team or organization, move forward wisely in the face of available alternatives? In this article, I explore the implications of Arrow’s Theorem for strategic decision-making and offer insights on how to navigate the paradox of choice.
You are sitting in a situation room with a team of important colleagues in your organization having been tasked to reach consensus on some strategic matter. Examples of the latter could be whether to terminate a product line, develop a strategy to adopt AI, pursuing a sustainable business practice, or choosing between project alternatives based on a set of criteria your project management office has developed over the years.
Suddenly, someone comes up with a clever selection scheme to reach consensus: “Let us each grab a sheet of paper and rank the available options on the table based on our individual preferences. The most highly ranked is the winner.” Everybody applauds in exhilaration, being relieved that they will finally vote on such a strategic matter and reach consensus in the end based on their individual preferences.
Despite our best efforts to devise intelligent and engaging ranking schemes for voting endeavors, such attempts are ultimately futile. Kenneth Arrow, who was awarded the Nobel Prize in Economics for his many contributions to the field, demonstrated this futility through his groundbreaking work on what is now known as Arrow’s Impossibility Theorem (AIT). This fascinating theorem shows that no ranked voting system can satisfy all the criteria for fairness, no matter how well-designed it may be.
Before I describe the theorem and its fairness criteria, I’d like to emphasize why AIT matters.
This Shall Make You Smarter
Do not squander your time and energy in the pursuit of a clever ranking scheme for selecting alternatives, options, scenarios, product pathways, or even job candidates. Arrow demonstrated through his groundbreaking work that no individual-preference ranking scheme can be truly fair when applied to a larger group, be it a team, organization, or community. The quest for the perfect ranking system is a futile one; instead, focus on making informed and strategic decisions with the knowledge you have at hand.
Particularly in matters of strategy, it is crucial to take decisive action and make informed choices. By mobilizing your resources, capabilities, action plans, policies, and attempts towards a pivotal alternative or two, you can focus your efforts and move forward with confidence. While Arrow’s Impossibility Theorem implies that not everyone will be satisfied with the outcome, your focus must remain on the bigger picture and the overall goals of your team, organization, or community. By making strategic choices with the big picture in mind, you can navigate the complexities of decision making and achieve success.
While it is true that no ranking scheme can be perfect, it may still be worthwhile to try to find a a way to make choices that works well most of the time. Ultimately, the focus should be on the whole rather than individual elements when it comes to strategy, and organizations need to pay heed to this observation. However, it is important to consider the potential consequences of leaving some individuals unhappy and try to mitigate them if possible. This is where strategy formulation, management, execution, reformulation, etc., enters the complex world of human behavior, art, systems dynamics, chaos theory, and the rest of the entries in the panoply of human thought.
A Closer Look at the Theorem
AIT — a mathematical result of social choice theory and welfare economics — states that when individuals who will be voting on some matter have three or more distinct alternatives, no voting scheme based on ranking can possibly convert the ranked individual preferences to those of the group in a fair way. The fair way is determined by the satisfaction of all of the following criteria:
Non-dictatorship: No single individual has the power to determine the group's preferences (otherwise, what's the point of voting).
Unrestricted preferences: An individual can present any set of preferences.
Pareto efficiency: If each individual in the group prefers one alternative over the other, than the whole group should also prefer that alternative.
Social ordering: There's a complete an transitive ranking of the alternatives.
Independence of irrelevant alternatives (IIA): The ranking of other alternatives should not affect the group's overall preference between any two alternatives.
The last two criteria — social ordering and IIA — deserve to be accompanied by an example.
Social ordering
Your team of five is trying to decide on the priorities for your department's next strategic program in data and analytics. They have three options: launching a new data product, expanding into new market segments, or improving their existing data product lines. Each team member has their own individual preferences, as follows:
The social ordering (really more of a consequence than a criterion) requires that the team's collective preference between any two options should be determined by aggregating the individual preferences between those two options, regardless of how they rank the third option. For example, if most team members prefer launching a new product over expanding into new markets, and most team members prefer expanding into new markets over improving their existing product line, then the social ordering should be launching a new product > expanding into new markets > improving their existing product line.
However, this may not be possible if there is a cycle of preferences among the team members, as shown in this example. In the table above, for example, there is no consistent way to rank the options based on the individual preferences. This is clear a violation of the social ordering criterion and an illustration of AIT, which we often see in many situations, far more complex than the above, in our business ecosystems.
Note that even if you swap Barb's first and second preferences, the theorem's impossibility result still applies — it does not matter how your rank individual preferences (say, to make one appear in majority) — since no swapping or clever scheming will satisfy the fairness criteria of the theorem.
Independence of irrelevant alternatives (IIA)
Let's continue with the example in the table above. Imagine that after some deliberations over happy hour, the team decides that launching a new data product is the department's top priority, followed by expanding into new markets, followed by improving the existing data product lines.
Now, suppose a fourth option is added: hiring new sales executives. This new option is irrelevant to the team's initial choice between launching a new product, expanding into new markets, and improving their existing product line. So, according to the IIA principle, their choice should remain the same and they should still prioritize launching a new data product over expanding into new markets and improving their existing product line.
However, if the addition of the fourth option causes the team to change their mind and prioritize expanding into new markets over launching a new product, then this would be a violation of the IIA principle. The fourth option, being an irrelevant alternative, should not have affected their initial choice between launching a new product, expanding into new markets, and improving their existing product line.
A Word on the Alternatives
The theorem requires three or more distinct alternatives because if there are only two alternatives, then the theorem's conditions can be satisfied by a simple majority rule. A majority rule satisfies all five fairness criteria. Using the example above with the five team members, but only two preferences: launching a new product vs. expanding into new markets, we have:
Non-dictatorship: The team's collective preference is not determined by a single team member's preference, but by aggregating the preferences of all team members.
Unrestricted preferences: The team can have any individual preferences over the two alternatives, such as preferring one over the other, or being indifferent between them.
Pareto efficiency: If all team members prefer one alternative over the other, then the team's collective preference as a group should also reflect that. For example, if all team members prefer launching a new product over expanding into new markets, then the team should also prioritize launching a new product over expanding into new markets.
Social ordering: If 3/5 team members prefer launching a new product, then the team's social ordering will be launching a new product > expanding into new markets.
Independence of irrelevant alternatives: The team's collective preference between the two alternatives should not be affected by the presence or absence of any other irrelevant alternatives. For example, if the team decides to prioritize launching a new product over expanding into new markets, then adding or removing the option of improving their existing product line should not change their decision.
Thus, with two alternatives, AIT does not apply since the criteria above are easily satisfied by a majority vote.
If the three or more alternatives are not distinct, they are not considered in the theorem because they do not represent different options for the collective choice. If two alternatives are identical in every aspect, then they can be treated as one alternative without affecting the preferences of the individuals or their group. Eliminate redundancy, since it doesn't really impact group decision!
Ordinal vs. Cardinal Alternative Selection
The theorem applies to ranked (ordinal) voting schemes, not to cardinal ones. In ranked (preferential) schemes, individuals can order their preferences strictly to show their ranks. In a cardinal scheme, individuals give each option an independent assessment (e.g., a rating grade from a 0-to-10 scale). Thus, unlike the ranked scheme, an individual here can give the same score to two alternatives.
The theorem must apply to ranked schemes since individuals provide less information on their overall preferences versus in a cardinal scheme where they let us know of way more information as to what they prefer equally or otherwise. Thus, in a cardinal system, group consensus-building becomes easier due to the inherent flexibility in individual preferences.
While cardinal schemes looks like a bed of roses, the reality is that even in simpler organizations it is very hard to be employed in practical ways. Diversity in opinion and agendas, power gaps, the multiplicity of interactions, the many symbolic and economic variables impacting individuals and teams, cultural performance, and many more factors make cardinal schemes quite impractical to swiftly act on some strategic matter. On the other hand, Arrow's result makes the bed of roses sound utopian. However, in practice, there is a way to act swiftly on strategic matters (as is often required) while being as fair as one can in choosing the winning option. Let's see what we can do in that respect.
Strategic Choices
In the context of strategy formulation and, really, on-the-spot adaptation as surprises enter your execution efforts, Arrow's Impossibility Theorem (AIT) suggests that there may be no good way to aggregate group preferences that will always satisfy the criteria we discussed above. Thus, any methods that can be introduced to combine individual preferences to make strategy a fairer process require some sacrifice be made. This is the reason why in any strategic endeavor, some stakeholders will come out exasperated and irked, some others happy and glorious. I'd argue that's a healthy sign of a swift strategic move forward! Conversely, if consensus is reached without contention on some "clever voting scheme", it's an ominous sign of strategy failure.
Some choice-selection schemes that come close to satisfying AIT criteria (save the independence of irrelevant alternatives, as I mentioned above) are:
Borda Count: This is a ranked voting method where individuals rank choices in order of preference. Points are assigned to each individual based on their ranking, with the highest-ranked choice receiving the most points. The choice with the most points wins.
Condorcet Method: This is another ranked voting method where individuals rank choices in order of preference. The winner is determined by comparing each choice to every other choice in head-to-head matchups. The choice that wins the most head-to-head matchups is the winner.
Approval Voting: In this method, individuals can vote for as many choices as they approve of. The choice with the most votes wins.
Range Voting: This is a method where individuals assign a score to each choice within a specified range (e.g., 0 to 10). The choice with the highest average score wins.
Majority Rule: This is a simple method where the choice with the most votes wins. It is also known as "first-past-the-post" or "winner-takes-it-all."
As an example, we can apply Borda Count to the table with three alternatives above. Any first-preference choice gets 3 points, any second-preference 2, and any first-preference 1 point. Adding them up, we get the following totals, resulting in launching the new product as the winning alternative:
Note that, in general, Borda count violates the independence of irrelevant alternatives criterion of the theorem if we introduce a fourth alternative. For instance, Ali may choose the fourth addition as his first preference and not even vote for his previously third preference (which would then get a score of zero in the Borda count).
Each of these methods has its own strengths and weaknesses, and none of them perfectly satisfy all of Arrow's Impossibility Theorem's criteria. However, they can be used in strategic management and, in particular, during strategic orientation (formulation) when teams need to make decisions based on the preferences of multiple team members.
Given the impossibility aspect of AIT, some individuals will come out of the situation room unhappy and perhaps filled with grudge, and some others victorious. Regardless of individual feelings, the overall wisdom has it that any organization would be at a severe disadvantage if it wastes time trying to come up with sophisticated choice-selection schemes versus acting on making strategic decisions swiftly. Strategy in this sense is the craft of making hard choices swiftly in order to move ahead in the game.
Happy Strategizing!