Game Theory Part 2
The Nash Equilibrium
At the end of our last article, we put an interesting spin on the iterated prisoner’s dilemma:
What if we have only 3 rounds to work with - we cooperate on the first 2, but then our opponent betrays us on the third one and we can’t get him back.
The unfortunate reality of this hypothetical is that, chances are, our opponent is going to betray us in the last round. If we can’t retaliate after that, why on Earth would our self-interested opponent want to cooperate, especially since betraying us would yield a greater reward? But that doesn’t necessarily mean we have to be the suckers. If we know that, given their incentives, our opponent is going to betray us in the last round, why should we still cooperate? If we’re going down, our opponent is coming down with us.
And so, we are also going to betray our opponent in the last round. But remember, our opponent is thinking the exact same thing, that their opponent (us) is going to betray them in the last round. Well if our opponent knows that we’re going to betray no matter what then they might as well betray us in the second round. Why? Because that move extracts the greatest rewards - betraying us in the second round is better than cooperating in the second round. But if we know that our opponent is going to make this move, then we, wanting to avoid being the suckers, are also going to betray in the second round.
The same happens for the very first round - our opponent knows we’re going to betray in rounds 2 and 3, and so they might as well betray us in the first round. Well, since we know this, we’re also going to betray in the first round. In short, adding a fixed number of iterations for the prisoner’s dilemma leads to both players employing a non-cooperative strategy, which is exactly what it sounds like - we’re at each other’s throats.
Nash Equilibrium
Let’s look at another game theory scenario. Say you and your friend are going mining and you both know that there are 2 clumps of silver and just 1 clump of gold in the area. Each of you can bring the equipment necessary to extract only one type of meta, either gold or silver. The clump of gold is more valuable than 2 silver clumps combined, but it’s a 2-man job to extract. The silver clumps, however, can be mined with just one miner.
Similar to prisoner’s dilemma, we have 4 formal outcomes:
Both Miner 1 and Miner 2 pick up their gold mining equipment and extract the clump of gold, which they can then sell for $1000 each
Miner 1 gets his gold equipment, but Miner 2 decides that he wants to mine some silver instead. Because Miner 1 can’t mine the gold on his own, and because Miner 2 is the only one with the proper silver-mining equipment, he sells both clumps of silver for $700
Miner 2 gets his gold equipment, but Miner 1 decides that he wants to mine some silver instead. Because Miner 2 can’t mine the gold on his own, and because Miner 1 is the only one with the proper silver-mining equipment, he sells both clumps of silver for $700
Both Miner 1 and Miner 2 pick up their silver-mining equipment and claim a clump, earning them each $350
Solving the prisoner’s dilemma (the original version, not the iterated one) required what’s known as a dominant strategy, which is the one that yields the greatest reward regardless of the opponent’s move. In the prisoner's dilemma, the dominant strategy is to betray, since no matter what your opponent does, you’ll be in a better position than if you had cooperated. While it works for prisoner’s dilemma, it won’t get you very far for this mining game. Why? Because the optimal move is completely dependent on whatever your opponent does.
Say you (Miner 1) knew that Miner 2 chose his gold-mining equipment. In this case, you should also fetch your gold-mining equipment, since earning $1000 for mining is better than earning $700 in a couple of silver clumps. But if you knew that Miner 2 was instead going with his silver-mining equipment, then that’s a completely different story - you too should pick up your silver-mining equipment since earning something ($350 for a clump of silver) is better than earning nothing from the clump of gold you’re unable to mine. This is all true from Miner 2’s perspective as well.
Now here is where we introduce the Nash Equilibrium. This describes a set of strategies such that no one wants to change them. Unlike dominant strategies, Nash equilibria are inherently stable - for it to be achieved, whatever Miner 1 is doing has to be optimal given what Miner 2 is doing and vice-versa. So how would we apply this to our scenario? Simple: we look at each of the four outcomes and decide whether or not any of the players can change their strategies to do better. There are actually 2 Nash equilibria in our scenario, namely outcomes (1) and (4)
Outcome (1): Both miners get $1000, which is the most money either of them can receive. There is literally nothing they can do to improve their situation
Outcome (4): Both miners get $350 each, but if either one of the miners makes the opposite move (picking up the gold-mining equipment), then they end up with nothing since they won’t be able to mine the gold. Hence, earning $350 is the best option
The Nash Equilibrium is much more dynamic than the dominant strategy since you can’t look at it from the perspective of only one player, you have to consider what both of them will do. This makes the Nash Equilibrium more cooperative in a sense, since the success of either of the players requires considering what the other one will do. In our next article, we’ll go through a more complicated version of the Nash Equilibrium and then tie in all this game theory with microeconomic theory.
Econ IRL
How does monetary policy (the control of the money supply) affect mergers and acquisitions (M&A), where the ownership of companies are transferred to or joined with other entities? This week’s paper asks just that and has 2 key findings. One, the likelihood of becoming an acquirer, that is, someone who’s buying a firm, decreases with a contractionary monetary policy, especially if it’s sudden. This is when a country’s central bank raises interest rates to reduce inflation, unlike expansionary monetary policy which seeks the opposite.
Being really precise here:
...we find that a one percentage point increase in the 1y Treasury rate reduces the likelihood of engaging in a M&A transaction within the following 4 quarters by 1.1 percentage points.
In other words, 1% increase in the interest rates = -1.1% in the likelihood of engaging in M&A transactions. The second finding concerns the type of firm more susceptible to engaging in these kinds of deals. More financially constrained companies, meaning ones who have lower profit margins and are valued less, experience a significant decrease in their likelihood of M&A transactions. There are 2 interesting hypotheses as to why this is the case, neither of which are mutually exclusive with the other.
The first explanation is that monetary policy affects a firm’s financial resources by altering the value of its assets (through either inflation or deflation) which then affects how much a firm is willing to borrow money to complete the merger or acquisition. Less financially constrained firms, almost by definition, can cope with this altering in asset value much easier than more financially constrained firms. The second explanation is more straightforward: firms with more market power are more likely to be financially unconstrained and may also want to engage in more M&A transactions to keep their dominant position.
‘Till next time
SoBasically