Games we play

Arpit Chaturvedi

(MBA, SCMHRD – 2011-13, MPA Cornell University)

In graduate school, there are two dominant concerns that occupy people’s minds, and these are the most common topics of discussions in friendly circles where people come together and endlessly strategize over these concerns. These two dominant concerns are – (a) the pursuit of getting a good job and (b) the pursuit of dating or coming into a relationship with a desirable partner. Having taken Prof. Kaushik Basu’s course in Game theory and having been part of endless strategizing sessions advising and helping friends about these two concerns (and grappling with these concerns myself), I decided to draw some insights from the discipline of game theory to resolve these perpetual and seemingly intractable issues. However, all my analysis and advice here should be taken with a grain of salt, as I am a single guy with no current job.

In the discipline of Game theory, a Nash equilibrium is a stage where having adopted a strategy, no player wants to deviate from it. The concept of Nash equilibrium applies well in both of the aforementioned concerns. To begin with, let me illustrate the concept with the easier example. Let us say there are two players in a game – a boy (B) and a girl (G). The boy has a crush on the girl and wants to enter into a relationship with her (this came could be played in the opposite direction too or in any other combination – boy-boy, girl-girl etc.). So, B has two strategies – pursue the girl (P) and not pursue the girl (P’) – shown in the first column of the matrix below. Similarly, the girl let us say has two strategies – whether to accept the proposal (A) or to reject the proposal (A’).

Let us look at the payoffs of this game:

          

A A’
P 8, 10 -5, 0
P’ 10, 10 -3, 0

In the above game, let us assume that if the boy decides to pursue (P) and the girl accepts (A), the boy gets a benefit of 8 utils and the girl gets the benefit of 10 utils (the boy could have got 10 utils but let us assume, pursuing has a two util cost attached to it, so the boy ends up with 8 utils having lost two utils in the cost of pursuing). In a scenario, where the boy decides to pursue (P), and the girl decides to reject or not accept (A’), then the boy gets a utility of -5 and the girl gets a utility of 0. In a situation, where the boy decides not to pursue (P’) and the girl decides to not accept (A’), the boy may feel some regret of being single, so the payoff for B is -3 (or it may even in some cases be 0, which does not change the game considerably). The girl, in such a case still gets a utility of 0. However, in case if the boy decides to not pursue (P’) and the girl decides to accept (A), then the utility for B, let us say is 10 and for G is 10. This is because the boy did not pursue and still got the girl, so he receives a benefit of 10 and the girl got a suitable partner so she gets a benefit of 10 again. This (the bottom left cell in the matrix) is the Nash Equilibrium because nobody would want to deviate from there. If the boy decides to move from P to P’, then he loses 2 utils (he comes from a benefit of 10 to a benefit of 8). If the girl decides to change from accept to reject (from a to A’), she goes from 10 utils to zero utils. For neither of them is it worthwhile to deviate from this strategy.

On the other hand, if the boy pursues (plays P )and the girl accepts (A) then the game is in the top left cell (PA), it is worthwhile for B (boy) to shift to P’ to pick up 2 extra utils. G (girl) is indifferent between P and P’ here as she gets 10 utils in either case. Similarly, if B plays P and G plays A’, it is worthwhile for B to play P’ to come from -5 to -3. Similarly, if B plays P’ and G plays A’, it is worthwhile for G to shift from A’ to A to pick up 10 utils instead of 0 utils. Therefore the only Nash equilibrium, given these payoffs is P’A.

One pre-condition of this game is that the interest of B should be communicated to G and that both should be rational players (this is where, the theory would not work out in the real world). Another assumption in this game is that this environment only has two players and there is a lack of choice for both the boy and the girl. However, often people get into perceptual locks where they are not able to see other choices or other choices are not available to them – or at least, there is a perception that other choices may not be available. In those cases, this model could work and this perceptual lock is not an uncommon phenomenon, even though the formalization of this concept may require more research.

This model can be applied to other marketing situations.

Now with the same payoff table, let us change the players. Instead of B, it is a student (S) applying for a job and instead of the girl (G), it is an employer, who is accepting or not accepting. Here, too, the interest of the student to work for this employer, should be known. The payoffs in these circumstances are more certain as the salary attached to the job can serve as a good proxy for the utility derived by the student from the job and the profit generated by the firm at a salary offered is known to the firm.

Let us say, the payoff matix is again like this:

      

A A’
P 8, 10 -5, 0
P’ 10, 10 -3, 0

In this case, however, we are sure that if the salary attached to the job is x and the profit derived by the firm is y, then under normal circumstances y would be greater than x, since the firm would naturally make more profit than the cost it is paying for hiring someone for the job. So the payoffs could look like this:

      

A A’
P x-2, y x-3, 0
P’ x, y 0, 0

Here again, the Nash equilibrium would be P’A.

Until now, the lesson from this analysis is that communicating interest without pursuing is the best strategy for the applicant.

In real life however, for each job position, there are more than one applicants. Let us say, there are two applicants for the job and they both decide to play P’ – i.e. not pursue for the job but simply communicate interest. In such a scenario, the firm would simply pick whoever increases the firm’s profits the most either by asking for the lower salary package or offering greater benefit by adding to the profits.

Let us say that the firm makes the two students S1 and S2, sit in different rooms and tells them that whoever quotes the lower salary package will be selected and if they both quote the same package then they both can be selected with a probability of ½. The maximum profit that the firm could derive from them is z. If both S1 and S2 quote a package, greater than z then none of them will get the job.  In this case, ideally both students should ask for a high package x such that x=z or just a little below z. If they both ask the same amount, they would end up with equal probabilities of being selected, albeit one will end up with 0 and the other will end up all of the amount.

On the other hand, if anyone asks for a lower package, that person would anyway be selected. Yet, what would happen here is that the two players would start lowering their quotes to a level below which they would not want to work. This is like Prof. Basu’s game of Traveller’s Dilemma. The firm will be able to hire the person with the lowest cost because the applicants would keep reducing their quotes.

Now let us consider a case where they can contribute an infinite sum to the firm’s profits. In this case, both should ask for an infinitely high package and should create the perception that they would be able to add an infinitely high number to the firm’s revenue thereby increasing the profit. In such a case, since the payoff for the firm in selecting either of the candidate is infinity, the firm would be indifferent to chose between either of them and would choose them based on simple probability.

In real life however, multiple firms are playing these games with multiple individuals and vice-versa. Yet, situations where a perceptual lock is created between a firm and a few candidates are not uncommon.

The success of the individual in being hired is also explained by game theory – this is when game theorists explain the phenomenon behind fads. Let us say in a population where an infinite number of people (firms) are seated adjacent to each other. Firm A has personal information that the applicant S1 can be a good employee – let us call it signal 1 (and let us call the feeling of not considering S1 as a good employee as signal 0). If firm A gets signal 1 and chooses S1, then it is upon firm B. If firm B gets signal 0 and chooses 0, then it is upon firm C. If firm C chooses 0, then all subsequent firms having seen that firm B & C chose 0, would keep choosing 0 regardless of their personal information. However, if firm B also chooses 1, then even if firm C feels 0, but sees that firm A and B chose 1, it would most likely chose 1 and not go with its own feeling or personal information and all subsequent firms would choose 1, regardless of what they feel. Hence, it is important for the employee to ensure that atleast firm A and firm B chose 1 and not zero or atleast firm B and C chose 1 and not 0. So giving the impression that you have atleast been selected by 2 earlier firms  (or actually being selected by atleast 2 firms) would play to the applicant’s advantage. Therefore, it is not unlikely to see that people who have some job offers, and who are able to communicate (perhaps not even in explicit terms) that they may have more job offers, tend to get more job offers as well.

Again, as mentioned earlier, all these advises should be taken with a grain of salt due to the reasons mentioned in the first paragraph that the writer does not particularly enjoy repeating!

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