“And then we have the economists who believe in game theory; arguing that we should disregard the health of our parents and grandparents!”
It was about six months into the covid-19 pandemic. I was bored and made the mistake of logging onto FaceMelt. Despite my best efforts I got a bit irritated although I wasn’t surprised. This may have been one of the oddest online arguments I saw on social media in the early days of the pandemic. What was strange to me was the anger I heard from people and how it cropped up all at once among a group of people I knew. They seemed to think that game theory was some sort of dishonest argument used by economists to justify treating people poorly. As an economist this caught me by surprise for two reasons. First, in the early days of the pandemic most economists were arguing that we had to protect people particularly in the face of the extreme uncertainty around both the transmissibility and fatality rates associated with the disease. The second reason it caught me by surprise was that game theory is really just a mathematical tool kit used for analyzing decision making and understanding likely outcomes based on people’s personal preferences and the situations they were in.
It was clear that there was some misunderstanding about what game theory actually is and why economists find it a useful tool. So what is game theory?
One of the common text books used in social science graduate programs for game theory is “Game Theory for Applied Economists” by Robert Gibbons. He defines game theory as “the study of multi-person decision problems”. This doesn’t really tell us much though. Perhaps taking a step back into how economists think about problems might actually clear things up a bit.
Classical economists in the 18th and 19th centuries were curious about how prices for goods emerge. Most of them, including Adam Smith and Karl Marx, tried to model prices as arising from objective value. The idea was that you could take a finished good and calculate the cost of all the inputs including labor, sum those input costs up, and the result would be the price of the item. In a sense, they thought of value as being an objective measure of the value of a good.
There were clearly problems with this model. First, people often put labor into things that others aren’t willing to pay for. Second, there are things that don’t have any inputs like labor that people value. An example of this are some natural resources like wilderness areas. Third, mathematicians and economists had been studying problems of risk and understood that often people were hurt more by losing something than they benefitted by gaining the same thing. This led to the idea of a utility function that mapped consumption of a good to a continous function and allowed certain behaviors to modeled mathematically.
In the late 19th century a mathematician named Leon Walras took a different approach to the question of value. He decided to consider two types of agents with respect to some good. The first was the producer. He built a simple model of the producers decision problem of how much to make and said that in general producers would want to maximize their profits. The goods they made would be limited by the technology and resources the producer had available to actually make the good. When he solved this optimization problem he got an upward sloping curve that showed as the price of a good went up the amount the producer would be willing to sell would also go up.
Next, Walras considered the problem of the buyer. He said that the buyer would have their own unique preferences and would seek to maximize their utility subject to those preferences. The buyer’s decision would be limited by the resources they had. When Walras solved that optimization problem he got a downward sloping line that showed that as prices decreased a person would want to buy more of an item. Where this downward sloping line intersected the producer’s upward sloping line would mark an “equilibrium point” where the system would settle automatically based on the buyers and sellers in the market.
This idea was a revolution in economics (in fact it is known today as the “Marginal Revolution”). The model showed that prices were not objective measures of value but were dependent on the relative values and preferences of agents in the market. The preferences and resources of the agent were the drivers of prices.
I breezed over some fairly restrictive assumptions in this model. One of them is that there are enough agents in the market that the actions of one person doesn’t impact the equilibrium price or quantity. This is a really important idea. If me deciding to buy or sell an item can move the market than the assumptions in the optimization problem don’t hold. A producer of mobile phones doesn’t really care specifically about me. They care about the aggregate decisions of all people in the market.
A brief example may help make this clearer. Imagine I own 100 shares of Tesla stock. I consider my preferences around risk and my estimate of future growth of Tesla and I decide that I would rather own Microsoft stock in order to maximize my personal benefit from owning equity in a company. The next day I can check the price of Tesla and Microsoft stock and if the prices look good to me I can sell Tesla and buy Microsoft. The prices I see for both equities can be reasonably assumed to be the prices I can sell and buy at. Now imagine my name is Elon Musk and I want to sell a portion of my stake in Tesla. I have to consider what news of my sale will do to the price of Tesla stock, and how the perception of my actions could impact the future capital raises that Tesla will have to do. I have to consider the actions and beliefs of others as part of my decision making process.
This second case is the type of decision making analysis that motivates game theory. What is the optimal choice when I have to take into account both my decisions, and the decisions of other people involved in the process?
The first formal paper in game theory was written by John von Neumann in 1928 where he proved the existence of equilibrium in a type of game. This was followed by a book on games written with Morgenstern. Later, John Nash proved the existence of a type of equilibrium in non-cooperative games that showed that in certain situations a system would reach a point where no one would benefit by changing their strategy. This type of equilibrium is a similar outcome to the market equilibrium and economists quickly saw that it could be used to solve problems in decision making that are frequently studied by economists but that don’t meet all of the assumptions that a market model needs to reach an efficient outcome.
So how are games actually modeled? In general you define a game with four elements. Every game must have players. You must define what information is available to the players in the game. The actions a player can take must be mapped out and you must define what the potential rewards for those actions are. Once you have those elements defined you can try to determine if there is an equilibrium point that the players will get to.
Games can take many different forms. Infinite games go on forever. Finite games, as the name sounds, have a defined end. In some games the players have complete information. In others they don’t. In some games the players are trying to cooperate with each other. In other games they are competing against each other. In some games both players make their decisions at the same time. In others each player takes a turn. This is important as many people’s perception of game theory is limited to a passing reference towards the end of an introductory economics course where they are shown a simple game like prisoner’s dilemna and think that represents the whole of game theory.
How are games practically applied in economics? They are often used in market design. For example, Hal Varian is the chief economist at Google and he was credited with designing the ad auction system that Google uses. Other economists have used game theory to design auctions of bandwidth for government agencies. Some use game theory to explore how economic agents learn over time. Have you ever been on an overbooked flight and the person at the counter starts offering money and credits for people to accept rebooking? Economists have been involved in designing those processes for airlines.
At the end of the day game theory is simply a mathematical field used to determine the best decision you can make given your goals, your beliefs, and the actions available to you when you have to take into account the likely behavior of other decision makers.