Hi Everyone,
I will continue my game theory series with the battle of the sexes game. Battle of the sexes is one of three famous games discussed in game theory. The other two games are chicken game and prisoner’s dilemma. I have posts on prisoner’s dilemma and chicken game. These posts can be accessed in links at the end of this post.
What is battle of the sexes?
Battle of the sexes game can be described as a game of cooperation and/or coordination. It is typically explained as a two-player game. The battle of the sexes game is often expressed as a game between a man and a woman. Both the man and the woman are better off by cooperating and/or coordinating their activities. The man has a preference for one activity and the woman has preference for another activity but they would both prefer to do any activity together as they value each other’s company.
Analysing the battle of the sexes game
The battle of the sexes is best analysed using the normal form game (normal form games are explained in the introduction to game theory post, the link to this post can be found at the end of this post). The normal game is presented in the table below.
Note: The first number is for the row player (Man) and the second is for the column player (Woman)
From the matrix above we can see that the man prefers going to the cinema than playing crazy golf. We can also see that the woman prefers playing crazy golf than going to the cinema. The matrix also tells us that the man and the woman get more enjoyment from doing an activity together either playing crazy golf or going to the cinema than doing a favourite activity by themselves.
Battle of the sexes does not have a dominant strategy for either the man’s or the woman’s favourite activity (dominant strategy is also explained in the introduction to game theory post). Instead, we have what is called a mixed equilibrium. Therefore, in theory all four outcomes are possible. In reality, only three outcomes should ever occur and hopefully only two outcomes should occur on a regular basis. The two outcomes are playing crazy golf together or going to the cinema together. The third outcome is each of them doing their favourite activity by themselves. The fourth outcome that is possible, in theory, is that both, the man and woman, do the activity they least like by themselves; an example of terrible communication and a bad outcome for both of them.
How to successfully play battle of the sexes
The two best outcomes, as mentioned earlier, are outcomes that where the man and woman are together. One of those outcomes favours the man (going to the cinema) and one of those outcome favours the woman (playing crazy golf). Which of these outcomes should be reached? If this is a one-off occasion the answer is not clear, either crazy golf or cinema with an equal probability (this equal probability is based on the payoffs shown in the matrix, different numbers could produce different probabilities).
Fortunately, battle of the sexes is often a repeated game (people go out with each other more than once, battle of the sexes situation typical does occur on first dates). If the game is repeated, a logical outcome would be to alternate between favourite activities. The matrix would indicates that a 50-50 split between going to the cinema and playing crazy golf would most likely be agreeable to both parties.
Real life examples of battle of the sexes
The battle of the sexes game can be applied to many real life events where cooperation achieves a better overall outcome. There are many ways to do things. Different people will have different ideas about how something should be done. If everyone just does what they think is best in the absence of any cooperation, a less than optimal outcome is quite likely.
Let’s take the example of filing papers into folders in the office. Imagine there are two types of hole-punchers, a three-hole puncher and a two-hole puncher. Two people may disagree regarding which hole-puncher to use. If they both use the three-hole puncher or if they both use the two-hole puncher, papers can be filed away together with ease. If they use different hole punchers, the papers cannot be filed together and will required different folders or re-hole punching as papers with three holes cannot be combined with papers with two holes.
Another more sophisticated example is the use of software. Imagine there are two firms (Firm A and Firm B) in the same industry deciding to update their computer software. These firms work together on a regular basis. Firm A likes software XYZ and Firm B likes software ABC. Unfortunately, the two software packages are not compatible with each other. If Firm A goes with software XYZ and Firm B goes with software ABC. The two firms will not be able to work well together anymore. If both forms adopted the same software, both firms can continue working together efficiently. The normal game for the two firms can be presented in the table below.
Firm A has been assumed to be the smaller firm of the two firms and would be hurt more from not adopting the same software as Firm B. This gives Firm B a stronger bargaining position in regards to agreeing on what software to adopt. This does not guarantee that Software ABC (Firm’s preferred software) will be adopted. Firm A, stands to lose out by a higher a percentage, if firms cooperate to both adopt ABC instead of XYZ. A mixed equilibrium approach would need to be investigated to determine which software is most likely to be adopted.
I will dedicate a post to explain mixed equilibrium as it can affect both the chicken and battle of the sexes games.
Conclusion
Battle of the sexes game is relevant in many areas. It can be seen as a nicer game than both chicken game and prisoner’ dilemma as cooperation and coordination is encouraged rather than competition. Battle of the sexes games are more likely to reach better outcomes if repeated. The players can make an arrangement to take turns regarding which outcomes to adopt. For example, go to the cinema one week and play crazy golf the next week.
I hope you enjoyed this post. I will be investigating game theory in regards to the Steemit bots in my next post. I will be looking at the real life application of game theory in a number of different areas. I will also explain the rather more complex ideas relating to mixed equilibrium which is relevant to battle of the sexes, chicken game as well many other types of games.
For more information on game theory, you can access my introduction post using the link below:
You can access my post on prisoner’s dilemma using the link below:
You can also access my post on chicken game using the link below:
