Game Theory and Office Politics: Coalitions

Office politics are one of these topics that no one wants to talk about. I’ve met people claiming that in their organization, politics do not exist. Others lament that politics are the root of all evil in their workplace. I think that politics are maybe sometimes unpleasant, especially when they negatively impact your work, but no one is helped by denying or demonizing their presence. Furthermore, it seems to me that their existence is only natural and that every human endeavor with more than two participants will necessarily bring forward some form of political behavior. It is only when politics lead to destructive decisions that we need to be worried.

To study office politics, when they are problematic, and how organizations can deal with them, game theory provides an interesting framework.

I don’t want to fall into the trap of name-dropping a mathematical framework just to lend a ring of authority to a pseudoscientific theory (looking at you, Simon Sinek). I am by no means an expert in the study of Game Theory or the domain of organizational sociology. Trying to connect the two, I am also going to handwave away technical details that some may find important. I have added links to some more technical material in [square brackets] at various places.

Game Theory is the Study of Strategic Behavior

Game Theory is a framework that studies the behavior of purely rational agents in strategic situations. The key idea is to model the strategic situation as a game in which every actor can choose from several strategies and then receives a reward depending on their strategy and depending on the choices of all other agents.

A company is an interesting setting for this framework as it is both an organization with certain shared objectives but also a collection of individuals with their contexts, aspirations, fears, and priorities. The “reward” that an individual receives in a given situation is not necessarily monetary, it could be about the ability to spend time with their family, status, future career opportunities, or social interactions with their colleagues.

Nevertheless, money plays an interesting role because it is one of the tools that help align the individual’s objectives with the company’s objectives: If the company does well economically, there is a higher chance of economic reward for all of the individuals working at the company. However, not all individuals benefit equally!

Let us consider the following slightly silly example: A situation in which four individuals are asked to vote (without knowing the choices of the other participants) on two competing proposals.

  • Plan red: Build a web app using React
  • Plan green: Build the same web app using Angular

Let us assume that the app will be moderately successful in both cases. However, with React it will be a bit cheaper because the team can reuse some existing code, increasing the monetary reward for the entire team when plan red is selected. Tom and Judy are React developers and would benefit more from selecting React as they would receive more praise. However, Tom is also worried about his work-life balance and does not want to be paged at 2 am as one of the only experts.

Anne and Cristopher are experts in Angular and would benefit more from plan green, relatively speaking. Anne is leading the team and gets the tie-breaking vote.

Let us put some numbers on this [as it is often appropriate to model individual preferences as utilities]: Let us assume that picking red gives everyone 5 points and picking green 4 points based on the organization’s performance. Every “team” gets two points if their preferred framework is picked, except Tom who only receives one.

If we plot all of the possible votes in this scenario, the solution is relatively simple: It is a [weakly] dominant strategy for each participant to vote for their preferred framework. As Anne has the tie-breaking vote, in this scenario option green is selected.

Option green is not the option that the organization prefers and also not the option that maximizes the sum of everyone’s utilities [the price of anarchy], however, in this scenario, the misaligned incentives have nothing to do with politics and can be fully explained by the fact that the individual incentives carried a higher weight in the decision than the organizational incentive.

Strategic Situations Can Be More Complex

Let us modify the scenario a little bit: Let us assume that Christopher is indifferent about the choice of framework. Let us further assume that there is a downside to being the only person opposing the boss: If the entire team votes against her preference, Anne is overruled. But if one individual votes red without the support of anyone else on the team, that individual will experience negative repercussions for their future career development. Let us assume that this is represented by a penalty of two points:

This situation is much less straightforward. Neither Tom nor Judy nor Christopher have a dominant strategy: Tom would prefer red. However, if he knew that the other two were planning to vote green, he would be better off voting green as well.

Overruling Anne forms a Nash Equilibrium: If you start from the situation in which Anne is overruled, none of the actors has an incentive to unilaterally deviate from their vote.

The Nash Equilibrium is a useful concept for studying strategic situations as it provides a sense of how a strategic situation can be resolved with some degree of stability. All actors could be said to be somewhat happy in a Nash Equilibrium as they could not have improved their situation by individually taking a different decision. For example, the solution where only Tom votes red is not a Nash Equilibrium: Knowing the outcome, Tom would have preferred to change his vote.

With that observation in mind, note that there is a second [pure] Nash Equilibrium in this game: The solution in which everyone votes green. This is because every individual actor that might have preferred red would get punished for deviating unilaterally.

Coalitions

In all of the above examples, we have assumed that each actor makes an individual choice: The team gets together, everyone raises their hand for red or green and then they leave. That is of course not how decisions are normally made.

In reality, it would be possible for Tom, Judy, and Christopher to align their votes beforehand. Here we get into the territory of political behavior! Note, however, that in this constructed scenario, the political move assures a better outcome for the organization!

Does Game Theory provide a tool to think about games in which coalitions can be formed? Indeed, it does: These games are studied as a part of Cooperative Game Theory (CGT).

In cooperative games, not only can actors collude, but they can also arrange side payments: As Judy benefits disproportionately from plan red, she might compensate Anne or Christopher in some way that was not part of the original game. The central questions studied in CGT are what coalitions form in a given setting and how actors can arrange fair side payments.

I will ignore the possibility that the payoff of a coalition may depend on the other actor’s actions [partition function games] and instead transform the game described above into a cooperative game as follows: A coalition can choose their combined actions to maximize over the worst possible response from the other actors [minimax].

This leads to a central element in the study of cooperative games: The characteristic function v which, for any subset of actors S, provides the payoff for a coalition that includes all actors of S. The coalition that includes all participants is referred to as the grand coalition.

Given the definition of the transformed game above, the characteristic function assigns to every coalition of just one person the best they can do assuming that everyone else is out to get them: For all players except Anne this means voting green to protect themselves against the repercussions. For Anne it does not matter what action she selects, she will have to assume she will get overruled.

Any coalition of two people that does not include Anne can collude and vote red without fearing repercussions. However, they have to assume that the opposing coalition will overrule them. A coalition that includes Anne can dictate the outcome. Note, however, that any two-person coalition that includes Anne would prefer red over green, except for a coalition with Christopher where there is no clear preference.

Note how V(JA) > V(J) + V(A), giving an incentive to cooperate.

All coalitions of three players maximize their score when red is selected.

Finally, the value of the grand coalition is simply the maximal sum of utilities V(AJTC) = 23.

So we now know that the total value of this game is 23 [because the game is superadditive]. From here it would be interesting to study the side payments that the actors would need to agree with each other to enter into a coalition. In other words: How will they divide the bounty among themselves?

Without taking this too far, no player would enter into a coalition accepting less than what they could get on their own [individual rationality]. So Tom, Judy, and Christopher would each expect to get at least a payoff of 4 each.

Furthermore, note that Tom, Judy, and Christopher don’t need Anne in their coalition to force their preferred outcome – therefore a solution to this game will likely not include compensating Anne [compensating Anne is not in the stable set].

Conclusion

Much more can be said on the topic of Game Theory, of course. After all, it is a topic that some people spend their entire life studying! I am glossing over many details in the above explanations.

Note in particular, that I still assume perfect information and that there is no uncertainty about the outcomes of the choices at hand – a completely unrealistic assumption in practice that warrants a post on its own!

A common criticism of game-theoretic models is the rational actor assumption: Do the people in question correctly estimate what they know and what they do not know and do they act following these beliefs? We know that humans are fallible. I would argue that while we certainly cannot assume perfect rationality, there are ways to make the organizational incentives obvious so that the rationality assumption is a reasonable approximation.

The most important points that I wanted to make in this article are the following:

  • In any organization, it is completely normal that different people have diverging incentives and we should not be surprised if they act accordingly.
  • This can lead to choices being made that are not in the best interest of the organization. While this can be the result of political behavior, it is more likely a result of a suboptimal incentive structure that does not sufficiently reward people for aligning their actions with the organization’s objectives.
  • We have also seen that a part of what we consider politics is coalition building. Such coalitions can be detrimental – but they are also a very natural occurrence unless the incentives lead to a straightforward dominant strategy equilibrium.
  • Furthermore, such coalitions may even be beneficial and lead to better decisions from the organization’s point of view like in the example shown above.

As we have seen, political behavior can be positive or detrimental depending on the incentive structure. This leads to the important question of how incentives should be structured to avoid negative consequences – a discussion that also deserves a follow-up.

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