I'm going to continue my very basic introduction to the branch of economics known as game theory that I started in my previous post, "Game On!". I started off discussing one of the most famous problems in game theory known as the "Prisoner's Dilemma". This classic game is known as a simultaneous game, i.e. a game where each player chooses their action without any knowledge of the actions chosen by the other players (i.e. players don't take turns), which are distinguished from sequential games, where one player chooses their action before the others choose their actions (as in the game Tic-Tac-Toe).
Another famous simultaneous game is Rock-Paper-Scissors. Simultaneous games are frequently depicted using a pay-off matrix, where each row describes the strategy of one player and each column describes the strategy of the other player (e.g. with two players, the pay-off matrix is shown as a 2x2 table). The intersection of each row and column reveals the pay-off for each player. For example, the pay-off matrix in a two-player game of Rock-Paper-Scissors (remember, Paper beats Rock, Rock beats Scissors, and Scissors beats Paper) would look like this:
The game of Rock-Paper-Scissors is also a good example of what is called a zero-sum game, which means that the only way that one player wins is if the other player loses (note that when you add up the pay-offs for Players A and B in each cell of the matrix above, the sum is zero). It is also called a non-cooperative game because two players (at least in this example) are competing against each other.
There are a couple of additional terms that I should define. As I mentioned in my first post ("Game On!"), the basic building blocks common to all games are the players, their preferences, the available strategies (which depend upon their preferences), and how these strategies affect the outcome of the game itself (which lead to a pay-off). Players use a pure strategy when they select only one strategy (for example, in Rock-Paper-Scissors, Player A could always choose "Rock" therefore guaranteeing a win 1/3 of the time), whereas they use a mixed strategy when they randomly select their responses (for example, Player A could choose Rock 2/3 of the time and Scissors 1/3 of the time). We will briefly discuss mixed strategies in an upcoming post that involves Premier League soccer and penalty kicks.
In every game, each player has a best response, i.e. the best strategic choice given a belief about what the other player will do. A dominant strategy is a strategy that is a best response to every strategy that the other player can choose. Finally, as discussed in our last post, the Nash equilibrium (named after the mathematician John Nash), occurs when no one can gain a higher pay-off by deviating from their best move.
There are three other classic 2x2 simultaneous games known as "Battle of the Sexes", "Chicken" (also known as Hawk-Dove and Snowdrift), and "Stag Hunt". All three of these games are known as coordination games, a type of simultaneous game in which a player earns a higher pay-off if they choose the same response as the other player. I will show a typical pay-off matrix for each of the three games and then briefly explain each game. Player A's choices are shown in the Rows, while Player B's choices are shown in the Columns. Similarly, Player A's pay-offs are the first number in each cell of the matrix, while Player B's pay-offs are the second number. The units for the pay-offs in these three games are utility units, or utils.
I am going to avoid the gender stereotypes that were part of the originally described "Battle of the Sexes" game. Ann and Bob are planning to meet up in the evening for a fun event. Ann likes to go to the Ballet, while Bob likes to go to the Opera. Both would prefer to attend the event together, rather than going to the event alone. If they cannot communicate, where should they go? If they both meet up at the Ballet, Ann will be the happiest (pay-off of five utils), though Bob will be happy enough that they went to the event together. Conversely, if they both meet up at the Opera, Bob will be the happiest, but Ann will still be happy too. Neither will be happy if they show up at different events. There are two Nash equilibria in this example - both going to the ballet or both going to the opera.
The "Battle of the Sexes" game illustrates the importance of standardization in business. Let's say both Ann and Bob are surgeons. The hospital would like to minimize costs, so they are standardizing the type of equipment used in the operating room. Ann prefers the surgical implant from the vendor, Balletine, Inc. while Bob prefers the surgical implant from the vendor, Operative Solutions, Inc. Assuming that there is no difference in the quality of care provided, they are better off using the implants from the same company versus different companies (which the hospital won't allow anyway).
The game of "Chicken" is also known as Hawk-Dove and Snowdrift. The 1955 movie classic "Rebel Without a Cause" has a scene in which the characters play a form of "Chicken" (in this particular version, the two players each drive a car towards a cliff, with the winner being the one who jumps out of the car last). In the typical version in game theory (this is absolutely not a game in real life), two drivers drive towards each other at high speeds. One of the drivers must swerve, or they both collide (which can be deadly). However, the driver who refuses to swerve wins the game, while the driver who swerves is called a "chicken" (coward). The pay-offs are easy to understand. "Chicken" has been used to model the strategy of brinkmanship used during the Cold War and particularly during the Cuban Missile Crisis. Here the Nash equilibrium is for one player to swerve and the other to go straight (and vice versa).
From a game theoretic perspective, "Chicken" is known in the field of biology as "Hawk-Dove" (where it is used to explain certain evolutionary behaviors. The game is also known as "Snowdrift" in a different version. In this friendlier version, two drivers are stranded on a road with a large snowdrift blocking their way. They each have shovels and can either work together to remove the snow and be on their merry way, or they can choose not to shovel and remain stranded. Again, we are assuming that they have to choose simultaneously and neither knows what the other driver will do beforehand. The "Snowdrift" version of the game is a great illustration of the "free rider problem" and the "Tragedy of the Commons" (more on this later).
Finally, "Stag Hunt" describes the scenario where two hunters agree to meet in the morning to hunt either stag or hare. It takes both hunters working together to successfully hunt a stag, though they can hunt for hare by themselves. Obviously, there is a lot more meat on a stag than a hare. Hence the pay-offs here should also be self-explanatory. Again, there are two Nash equilibria (both hunters hunt Stag or both hunters hunt Hare). "Stag Hunt" was first described by the philosopher Jean-Jaques Rousseau during the Enlightenment, and the game is often seen as a useful analogy for thinking about cooperation, such as in international agreements on climate change.
There is no way that I can explain all the nuances of these seemingly simple, yet incredibly complex games in game theory. However, hopefully I have at least given you a taste of what game theory is about (and introduce you to some of its most important concepts) and how useful it can be for modeling strategic behavior in a variety of settings. With that in mind, next time we will talk about some useful applications of game theory in the real world.
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