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Mergers and Acquisitions: The Game

In an article from 2012, a game theorist describes how to make money on the stock markets by modeling a couple of key players in mergers and acquisitions. In his article, he refers to company A as the acquirer and company T is the target. He stresses the importance of three key players: index fund managers, company A’s management, and company T’s board of directors. The theoretical scenario is as  follows:

1. Prior to the announcement, company A was trading at $50 and company T was trading at $20.

2. It is announced that company A will buy all shares of company T, paying 1 share of A for every 2 shares of T.

3. The price of T now raises to $24 per share.

The trade to do here is to go long 2 shares of T and short a share of A, and if the merger goes through, you would be flat in positions with $2 extra in your pockets. The tricky part is figuring out what the probability is for the merger to be completed.

Enter game theory: what are the payouts for index fund managers, company A’s management, and company T’s board of directors, in each scenario?

Aaron Brown tells us that index fund managers sit still and go even regardless of the outcome, or they can sell T and buy A, betting that the merger will not go through, and the price of T goes back to it’s original $20 (it could drop lower due to a hit in investor confidence, but that is a minor issue).  Supposedly, mergers in which this strategy seems attractive are mergers in which you’d like to buy T and short A.

There are a couple of reason why I disagree with this, the first of which is that these index fund managers can do the exact opposite of Brown’s suggested strategy: they can mimic you and buy T and sell A, betting that the merger goes through. Unless this is not the case, index fund managers should play the exact same role as other merger arbitragers. The second reason is that if this “sell T buy A” strategy seems attractive to these index fund managers, it would imply that “buy T sell A” is particularly unattractive. That is, that we are expected to lose money betting on the merger going through.

Company A’s managers, according to Brown, are your ‘competitors’. Our “buy T sell A” puts us short A, and company A’s managers are long A. Thus, in some sense of the word ‘competition’, we are in competition with them. However, there are many other incentives for company A’s managers to make the merger work out, the strongest of which is the fact that company A is expected to be more efficient once acquiring company T, and so stock price would naturally go up in the long run. Looking at it in this perspective, A’s managers and us are on the same side – hoping for the merger to be completed. (mergers that fall through also reflect poorly on both companies, thus A’s managers REALLY want it to work)

Company T’s board of directors are our ‘allies’, since they and we alike are long T. However, in the perspective of the merger, that might change. Brown addresses some very important points: “They might be angling for board positions in the new company, or worried about criticism or shareholder lawsuits, or have emotional ties to T as an independent entity.”  While T’s board of directors may be afraid that stock price might fall under $20 in the ‘no deal’ case, they might prefer to other alternatives.

In my opinion, by far the most important players are the latter two. We can imagine a payoff matrix in which each company either Agrees or Disagrees to merge. The cases in which both agree and both disagree are simple – the companies merge or don’t merge. It is a little bit more complicated when one agrees and another disagrees. I am no expert on mergers, but I would guess that if A agreed to merge, and T does not, it reflects poorly on A, and vice versa. Once we know the expected payouts of each party in each of the 4 different scenarios, we can calculate the nash mixed strategy for the two, and have a guess of whether or not the merger will go through. From there, making edge is simply a matter of looking at the market implied probability that the merger succeeds, and trading against that with our model.

As Brown mentioned, complex models can lead to 100% accuracy in predictions in the past, while not being a good predictor of the future. What is nice about this game theoretical approach is that it is fairly simple, and does not make too many assumptions. However, I do believe that I’ve missed out on at least a couple of key players in the merger, and so a realistic payoff matrix would look more like a hypercube with side length 2 (accept/reject) in n-dimensional space, where n is the number of important players.

The final error in this strategy is that we are assuming that human beings are able to play a perfect mixed strategy. It is not clear to me that all players involved will calculate this payoff matrix (or consider each and every single one of the 2^n possibilities), and play the optimal strategy. Hypothetically, if a manager were extremely irrational and decides not to go through with the merger because he/she had a bad day, this strategy could fall flat on it’s face.

 

http://www.minyanville.com/businessmarkets/articles/game-theory-zero-sum-games-institutional/2/16/2012/id/39440?page=full

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