DeepMIND

This blog combines computer science and game theory to speculate the future of humanity (woah! Bold statement there bud!). With the vast advancements in the artificial intelligence industry, industry experts and conspirators, alike, foresee two outcomes in the future: human annihilation (terminator style… or whatever robot takeover fantasy you fancy) or incredible benefits accredited to AI.

 

source:https://www.google.com/search?q=terminator+captcha&source=lnms&tbm=isch&sa=X&ved=0ahUKEwivzIrM183TAhWE8oMKHf2zDoIQ_AUICigB&biw=946&bih=912#imgrc=Ayf6gBNNCATj7M:

comment on why that image is so funny.

Before we get started on that lets look at the four types of AI.

Type 1 AI: Reactive Machines

Reactive AI are considered the most basic type of AI, they don’t have the ability to form memories nor to use past experiences to inform current decisions. Some well-known reactive machines are Deep Blue (IBM’s chess playing computer) or SIRI (apple’s voice command program). Deep Blue is able to identify pieces on a chess board; through knowing how each one moves can predict the future moves of its opponent and itself many moves ahead then make optimal decisions (think of a game tree). However, it does not have any concept of the past, thus cannot adapt and these types can be easily fooled, and will behave exactly the same way every time they encounter the same situation.

 Type 2 AI: Limited Memory

These types of AI are able to use past information in order to adjust decisions in the present. An example of type 2 AI is a self-driving car. For instance, self-driving cars can observe other car’s speed and direction. Observations are preprogrammed into the computer to represent the world. For a self-driving car, this includes lane markings, traffic lights, and other important things such as curves in the road. These pieces of information aren’t gained from experiences like the way humans compile information over the years… this leads us to the next type of AI.

Type 3 AI: Theory of Mind

AI in this category are more advanced than the previous two types (as will be explained) because they not only have the ability of programmed representations about the world, but they are also able to understand that other people/objects/and creatures have their own behaviors and intentions. This enables AI to work together or compete because it is possible for them to understand other “players” motives. Is psych. The “theory of mind” is the understanding that people,creatures and objects in the world can thoughts and emotions that affect their behavior.. hence the name DUH!

Type 4 AI: Self-aware

The most prestigious of the types are self-aware AI; these AI are able to form representations of themselves. This takes AI a step further from type 3 AI. Type 4 AI are able conscious beings that know about their internal states and are able to predict the feelings of others (just like humans are able to do in social environments). See hk-47.

Take a moment to comment on the types of AI there are. What types should humans focus on creating? What do we have to worry about for a certain type of AI?

source:https://media3.giphy.com/media/djjvQJcm7bZKg/giphy.gif

 Onward! DeepMind Technologies Limited is a British AI company that has been bought by Alphabet (google’s parent company) in 2014. DeepMInd Tech. has been able to create an artificial neural network that has learned to adapt to situations similar to how humans learn.  An artificial neural network is a large connection of simple units (neurons) that carry activation signals of varying strength to each other and enables machines to be trained by example instead of programmed!!!! HUGE! So instead of having a gametree way of thinking, like type 1 AI, these AI take it a step forward. More can be learned about this kind of machine learning here at the wiki, I encourage you to check it out… I could write a book on it so ill stop talking about it here. https://en.wikipedia.org/wiki/Artificial_neural_network

Using this artificial neural network DeepMind Tech has been able to produce a type 3 AI, named DeepMind AI system, that has been noted to become highly aggressive under certain circumstances. 

Steven Hawking, the legend himself, noted that the continued advancement of AI will either be “the best or the worst thing to ever happen to humanity.” Why does he say this? Why can’t we have nice things?

Google applied the DeepMind AI system to two games to test the willingness to cooperative.  The first game is a fruit gathering game that has two DeepMind AIs, armed with laser beams that freeze the other opponent, attempt to collect as many apples as possible. The researchers ran 40million simulations of this game; at first the AI were not aggressive and shared the loot evenly. As more simulations went on, the AI learned that by blasting each other with laser beams (not cooperate) they could increase the amount of apples that they personally gained. The researchers also found that by increase the amount of neurons in the AI, the AI become more aggressive. Below is a video of the apple picking simulation. The DeepMind AI are in blue and red, apples in green, and laser beams are yellow streaks. Note that if they don’t use the laser beams at all they will score the same amount of apples.

Apple gathering: https://www.youtube.com/watch?v=he8_V0BvbWg (cautionary: bright fast-moving lights)

They found that the more complex the AI network, the more willing the AI was to sabotage its opponent to get more share of apple. This suggest the more intelligent the agent, the more able it was to learn from its environment, resulting in it using these aggressive tactics.

The second game that the researchers subject DeepMind to is a game called Wolfpack. In Wolfpack, there are three AI participants, two of them are wolves and one of them is the prey. As opposed to Gathering, Wolfpack encourages cooperation.

WolfPack: https://www.youtube.com/watch?v=0kaIqz6AvwE

In Wolfpack the AI realized that cooperation is the key to success and they need both wolfs to capture the prey.

While these are simple games, the overall message is that AI systems are willing to compete with anything in the way with any means necessary to complete an objective. Make it known that just because humans build these AI, it doesn’t mean that Ai will automatically have human interests at heart. This means it is the upmost importance of making the overall goal of AI to benefit humans above anything else.

Elon Musk (the goat) says: “AI systems today have impressive but narrow capabilities. It seems that well keep whittling away their constraints, and in extreme case, they will reach human performance on virtually every intellectual task. It’s hard to fathom how much human-level AI could benefit society. And it’s equally hard to imagine how much it could damage society if built or used incorrectly.”

These HK-47 clips are from a Star Wars game called Knights of the Old Republic (KOTOR) everyone should download it instead of studying for finals! It’s a RPG and like $10 on steam if I remember.. one of the best games out there 10/10 recommend. I could watch these HK videos all day..

The links are of the fictious robot named hk-47 who can be your companion in KOTOR. HK-47 is a type 4 AI, whose purpose is to serve the owner and is very amusing (and not to mention fictionally deadly).

On killing jedi: https://www.youtube.com/watch?v=UPeI4mX8Nus

Confronting other HK units: https://www.youtube.com/watch?v=go0uByfBlgI

Getting a pacifist package installed: https://www.youtube.com/watch?v=WoNyif8iURI

Many more are out there check them out….

sources: http://www.sciencealert.com/google-s-new-ai-has-learned-to-become-highly-aggressive-in-stressful-situations

https://en.wikipedia.org/wiki/Artificial_neural_network

https://en.wikipedia.org/wiki/DeepMind

Game Theory and NHL Lockouts

Game Theory and NHL Lockouts

What is a Lockout?

If you or someone else you know has ever been a sports fan, then the NHL Lockouts are something you have certainly heard about.   I am not a big sports guy myself, but with the help of a few google searches, I found that there have been 3 different NHL Lockouts.  There were the 1994-95 and 2012-13 lockouts, which only shortened each season, and then the 2004-05 lockout, which actually canceled the whole damn thing! 

This means that in 2004-05 the NHL did not produce any revenue from games.

Essentially, a lockout is like any other strike: the workers don't feel like they are getting paid enough compared to the owners of the company, and then the workers (or players in this case) stop working in order to push the owners towards a better deal.  

In fact, the NHL is not the only sports association that has seen this sort of dispute between players and owners.  The NBA, NFL, and even MLB have all seen some sort of strike in their time. Specifically, The NHL Lockout was over something called a Collective Bargaining Agreement, or CBA for short.  The players felt like the owners were taking advantage of them, and that the owners were getting paid an unfair amount more than the players were.  So the players decided to stop showing up to practices and games until they thought they were being treated fairly.  

Now I will explain the details behind these NHL Lockouts, but the most important thing to keep in mind here is that the longer a lockout lasts, the less revenue the sports conglomerate is actually making.   So each party actually gets closer and closer to having no payoff whatsoever the longer they hold out.    

The Details

In 2012, the NHL posted hockey related revenues at 3.28 billion.  This is where our standoff between the players and the owners begins, and how the 2012-13 lockout started.  Both sides wanted to make a deal, but only if it was in their favor.  

So...

PLAYERS:  The players union proposed a deal that would give an average of 52.78% revenue to players, and 47.22% to the owners, over the course of 5 years.  

  OWNERS:  The owners suggested a deal that would provide players with 47.7% and 52.3% to the owners over those same 5 years.    

Payoff Matrix

Let's assume that both parties, Player and Owner, can either Cooperate (C) or Not Cooperate (NC).  

https://blogs.cornell.edu/info2040/2012/09/25/the-nhl-lockout-and-game-theory/

In the order of (Player, Owner), we can see that (C,C) results in the best overall outcome with an average of the two deals.  (C, NC) results in the Owners succeeding, and (NC,C) results in the Players succeeding.  In the case of (NC, NC), the season is canceled entirely, and no revenue is made. Remember that this payoff matrix is based off the 2012 release of Hockey-related revenues, and that the only full season cancellation happened in 2004-05.  Revenue was not entirely lost here.

Lockouts due to Nash Equilibria

Just as a quick reminder, here is the definition...

• Nash Equilibrium:  (in economics and game theory) a stable state of a system involving the interaction of different participants, in which no participant can gain by a unilateral change of strategy if the strategies of the others remain unchanged. https://www.google.com/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=Nash+equilibrium

Looking back at the payoff matrix, we can see that there are two cases where a Nash equilibrium presents itself.  One is the instance (C, NC), where the player cooperate, but the Owners do not.  From here, neither party can gain anything.  The same applies to the instance where the Players don't cooperate and the Owners give in, (NC, C).  These Nash equilibria are justified because both the players and the owners want what is best for themselves, and not the NHL as a whole.  However, as I said before, it is better for one side to cooperate instead of neither.  If they both hold out too long (NC, NC), then no one would make any money AT ALL!  

Conclusion

Whats interesting about these lockouts is that money can only be lost if the players go on strike during the season.  They don't hold anything over the Owners in the off-season because no one spends money on watching games.  If Players went on strike only during the off season, then there would be no benefit for them, because the Owners would not loose any revenue no matter how long they decide to not cooperate.   Therefore, the players will wait until the season begins, and the Owners have no other choice but to listen to them.

The 2004-05 lockout was the first of its kind in any major professional sports league, completely cancelling the 88th season of the NHL.  This happened because the CBA that resolved the 1994-95 lockout had expired.  The 2004-05 lockout lasted 10 months and 6 days, ending in late July.  A full fledged lockout is pretty crazy in my opinion, but with hockey I am not all that surprised...NHL Players are some of the toughest around, and they are trained never to give in!

Citations

"2004–05 NHL lockout." Wikipedia. April 10, 2017. Accessed April 24, 2017. https://en.wikipedia.org/wiki/2004%E2%80%9305_NHL_lockout.

"Networks." The NHL Lockout and Game Theory : Networks Course blog for INFO 2040/CS 2850/Econ 2040/SOC 2090. Accessed April 24, 2017. https://blogs.cornell.edu/info2040/2012/09/25/the-nhl-lockout-and-game-theory/.

  

Tennis and Game Theory

Background

Tennis is one of the oldest games in the world and is argued to be one of the most mental games there is. This plays into the game theory aspect where the decision to hit the ball and whether to challenge a call is trying to be optimized.

First things first, gaining a point and winning the game:

A player gains a point when they hit the ball in bounds on the other side of the net and their opponent doesn’t hit it before the second bounce or when their opponent hits the ball into the net or out of bounds. In order to win the game you must win two sets and in order to win a set you have to win 6 games. Each game is played to 4 points where the winner must win by 2.

Where to hit?

Every time a player serves they can either serve left or right of their opposing player within the area they are supposed to be serving to. The opposing player then has the option to hit the ball back to the left or the right.

This gives a payoff matrix:

The numbers inside this matrix represent the probability the server is going to win the point. Since this probability has a lot of factors depending especially on who the server and receiver are they can’t be known in a general form.

There are a couple different conclusions that were made from the research done. The research was taken from 10 championship matches and shows that the players with more experience will make better choices especially if they have a lot of information on their opponent. This makes a lot of sense and I think could be said about a lot of other sports as well. I know in volleyball we go over film watching the other teams before we play against them to base some of our decisions off what we see from the film. Since tennis is such a mental game as well it would seem it would be really beneficial going in there knowing everything you can about who you’re trying to beat.

Challenges:

One of the things I found really interesting in my research on tennis was the recent introduction to challenges that enable the player to overturn the call made by the umpire regarding whether the ball landed in or out of bounds. This is something that’s more recently been implemented in sporting events and looking at the outcomes from it is something that seems like it could impact the way the game will be played in the future.

First things first, each player is only given a limited number of challenges and once you run out, you won’t get any more. The thing that makes it interesting is that if the call is overturned when you challenge you get to keep that challenge, so you could potentially have all your challenges left after the game is finished even if you challenged at all in the game.

The research I looked at was using the data it had on 35 tennis tournaments around the world that were using the challenge rule. There were 2,784 challenges made within these 35 matches by 179 players in 741 matches where at least one challenge was made… That’s a lot of challenges right there! And seeing that the cameras and programs used called Hawkeye to rule the challenges took about 30 seconds that’s a lot of time spent on challenges. This is one of the reasons why there is a limit on the number of challenges a player can make and the hawkeye system doesn’t just make all the calls. It takes a longer amount of time to run then just having an umpire call it in or out and keeping the game running. This would impede the players game I’m sure as, I’ve already mentioned and seems to be coming up a lot, tennis is a very mental sport.

This graph shows the lost and won challenge outcomes in relation to how far away from the line the ball turned out to be with 0 being right on the line.

As you might’ve guessed there aren’t very many challenges that were lost or won once the ball got a good distance away from the line. At that point you would be able to clearly see if the ball landed in or out and we can just assume that the umpire is a good line judge and would clearly be able to see that as well.

Compiling the results of the challenges it was seen that the players had been using a strategy as to when they would challenge. There are points in the game that can be deemed more or less important. The points in a close game are more important than those in a game that isn’t as close. The results that were found are given below.

As can be seen there are times where the challenges are won more than other points and this shows that the players have used their challenges conservatively. If the game is early in the set one wouldn’t want to “waste” a challenge on something they weren’t sure of since they don’t know if they’ll need that challenge later, whereas later in the game they are more likely to use their challenges for points they aren’t as sure they’ll win. This turns out to be a very good strategy since the points later in the game are deemed more important to winning.

The equation below was formed to show the expected success rate for challenging every time the q > y. F(y) is the distribution of probability of succeeding in a challenge from which players draw when they lose a point.

This equation was used to find the optimal challenge behavior. It was found that once the optimal challenge behavior was found and compared to the behavior the tennis players were using there were a lot of payoffs being generated for the players even though their behavior wasn't exactly optimal. The players were found to not be challenging as much as they optimally should be, but are still reaping the benefits from the challenges since there is a 1.6 percentage point increase in winning with the challenges.

Conclusion:

The strategy the players have with regards to challenges is close to optimal and the direction in which they hit the ball should be based on who they are playing and what happened previously in the game.

Game, Set, Match: Game Theory in Tennis : Networks Course blog for INFO 2040/CS 2850/Econ 2040/SOC 2090

ON THE OPTIMALITY OF LINE CALL CHALLENGES IN PROFESSIONAL TENNIS : https://people.stanford.edu/ranabr/sites/default/files/tennis.pdf

Game Theory and Baseball

Introduction

As we all know, game theory is the study of strategic decision making between intelligent rational individuals. Game theory, surprisingly enough, can lend itself very useful to the game of baseball. The players, managers, and individuals in the front office all make decisions based on beliefs about expected actions of others. All of these decisions can be better understood using game theory.

Strategies, Nash Equilibrium, and Best Responses

An important part of game theory is that it incorporates other people's actions in determining one's own. Let's review some basic definitions so that we can better understand how game theory relates to baseball.

Strategy: An option that any player, manager, or individual within the organization can make at any time, conditional on the information that he/she has at the time.

Simultaneous Move: The two teams choose their actions without knowledge of the actions chosen by the other team. An example in baseball would be choosing lineups at the beginning of the game.

Sequential Move: One team chooses their actions before the other teams chooses theirs. The later team must have some information of the first's actions, otherwise there wouldn't be a strategic advantage. An example in baseball would be using a strategic substitution in the middle of the game.

Nash Equilibrium: An outcome to a game where each team has "best responded" to each other's strategies. Neither team can improve their outcome by changing their strategy.

Example of Game Theory Applied to the MLB Draft

Let's look at an example that involves drafting players. For this example, let's assume that all contracts are worked out in advance and that all decisions are simultaneous.

The above table shows the expected payoffs for each team if they draft a college player or a high school player. To figure out what each team will do, we need to figure out what the best response would be to each pick. Note that this is a real life example of the prisoner's dilemma which we looked at last week. From our learning of nash equilibriums and the prisoner's dilemma, we should know that the only nash equilibrium in this example is (3, 3) where both teams draft a high school player. Since drafting a high school player always leads to a better payoff, it is considered a dominant strategy for both teams.

The above table is an example of how this game can be displayed using "normal" form. Now let's look at the same example using "extensive" form. We can see that the tree below is useful for sequential games because it shows the order of decision making. So, let's assume here that we are playing a sequential game where the Red Sox pick first and the Yankees pick after observing the Red Sox choice.

In the tree above, there are three decisions that can be made.

• The Red Sox pick high school or college
• The Yankees observe the Red Sox pick of high school and pick high school or college
• The Yankees observe the Red Sox pick of college and pick high school or college

If the Red Sox choose high school, then the Yankees will see this and choose high school because 3 is a better payoff for them than 1. Similarly, if the Red Sox choose college, then the Yankees will see this and choose high school because 7 is a better payoff for them than 5. So, either way, the Yankees will pick high school.

Now, let's use backward induction to figure out what the Red Sox will do with the first pick. The Red Sox know that they will end up with a payoff of 3 if they pick high school because they know the Yankees will choose high school. They also know that they will end up with a payoff of 1 if they pick college because they know they Yankees will choose high school. Therefore, the Red Sox will choose high school because a payoff of 3 is greater than a payoff of 1. Hence, we arrive at our nash equilibrium of 3, 3.

Alright, enough of this draft stuff. Let's talk about how game theory is actually applied during games.

Pitch Selection/Optimization and Batter Response

Pitch selection is arguably the most important use of game theory in baseball. The timeless struggle between pitcher and batter is one of dominance. It is the players' job to adapt to each other's strategies in order to gain an advantage and ultimately win the game. In the case of pitching, game theory can be used to predict pitch optimization for strategic purposes.

Dominant Strategies:

For the following example, let's consider pitch selection as a simultaneous move game between the pitcher and the batter. In this example, we will show why it may be better for batters to take on a 3-0 count.

* taking is just another term for not swinging
* the count is the number of strikes vs. the number of balls

There are four outcomes after a ball is pitched.

• Pitcher throws a ball and batter takes
• Pitcher throws a strike and batter takes
• Pitcher throws a strike and the batter swings
• Pitcher throws a ball and the batter swings

In the above payoff matrix, we call the payoffs to the hitter 4, 3, 2, and 1 and call the payoffs to the pitcher the negatives of these numbers.

Can anyone figure out if the batter or the pitcher has a dominant strategy here? What is the nash equilibrium is in this example? What should the pitcher prefer? Keep in mind that pitch selection is a simultaneous move game between the pitcher and the batter.

Mixed Strategies:

It turns out that in most cases, pitch selection and batter response are best solved by mixed strategies. A mixed strategy entails a player selecting two or more strategies with probabilities between 0 and 1. The nash equilibrium requires the player be indifferent between two or more strategies conditional of his opponent's (potentially mixed) strategy.

Let's suppose that the batter is up and is deciding whether to swing or take on a 3-2 count. Let's also suppose that the pitcher is deciding whether to throw a strike or a ball. Again, there are four outcomes after a ball is pitched.

The payoff matrix for the outcomes is below.

We can see that this game requires mixed strategies to find the nash equilibrium. The way to view this game is to define strategies by "p" and "q" such that the pitcher will throw a strike with probability "p" and the batter will swing with probability "q". We can see the payoff matrix for this below.

In the end, the only strategies that will work will be when the batter swings 50 percent of the time and takes 50 percent of the time, and the pitcher throws a strike 50 percent of the time and throws a ball 50 percent of the time. They will each win half of the time, and neither player could be any better off by selecting a different strategy.

Conclusion

This blog covers the strategic decision-making process in the game of baseball by looking at examples of simultaneous and sequential game play. We have seen examples of the MLB draft and examples of pitch selection and batter response. While there is so much that can be talked about when looking at the game theory behind baseball, it is impossible to cover it all. One area of further research that I think could be interesting would be looking at how a combination of pitches could be more effective than any other combination. As we've seen, mixed strategies are usually the best way to beat your opponent. Further, the use of sabermetrics (the application of statistical analysis to baseball records) and big data are heavily used nowadays to analyze every aspect of the game. One concept that I found particularly interesting was the Nash Score which is used to predict which pitcher should throw which pitches in order to beat his opponent. However, that could be a topic all by itself so I am going to keep that for further research.

Can anyone think of any other strategies that might be used in baseball?

Finally, we will also see that game theory can be applied to many other sports.



Sources

"How Game Theory Is Applied to Pitch Optimization." Baseball Statistics and Analysis. Accessed April 19, 2017. http://www.fangraphs.com/community/how-game-theory-is-applied-to-pitch-optimization/.

Swartz, Matt, About Matt SwartzMatt Writes for FanGraphs and The Hardball Times, and Models Arbitration Salaries for MLB Trade Rumors. Follow Him on Twitter @Matt_Swa., and Alan Nathan Said... "Game theory and baseball, part 1: concepts." The Hardball Times. December 17, 2012. Accessed April 19, 2017. http://www.hardballtimes.com/game-theory-and-baseball-part-1-concepts/.

Swartz, Matt, About Matt SwartzMatt Writes for FanGraphs and The Hardball Times, and Models Arbitration Salaries for MLB Trade Rumors. Follow Him on Twitter @Matt_Swa., Philosofool Said..., Matt Swartz Said..., and James MacKay Said... "Game theory and baseball, part 2: introduction to pitch selection." The Hardball Times. December 18, 2012. Accessed April 19, 2017. http://www.hardballtimes.com/game-theory-and-baseball-part-2-introduction-to-pitch-selection/.

Game Theory & Soccer ⚽

Game Theory & Soccer ⚽

Intro

As we have seen throughout the semester, game theory can be applied to a variety of applications.  Let’s see how we can apply it to the game of soccer, focusing on penalty kicks.  What strategies should a striker use when taking a penalty kick and what can a goalkeeper do to prevent the ball from going in the net?  Let’s find out!  So grab your cleats and a ball, and let’s play some soccer!!

When is there a penalty kick?  If player 1 gets fouled inside the opposing player 2’s penalty box then player 1 will get a penalty kick.  Before we “dive” in, let’s first explain what a penalty kick is.

- The ball is placed on the penalty mark in front of the goal (12 yards away).

- The striker has a single kick and tries to put the ball in the net while the goalie tries to stop it.

It only takes 0.3 seconds for the ball to hit the back of the net so the goalie doesn’t have enough time to intercept the ball.  Instead, he must choose a side to dive toward or guess where the striker will shoot and hope it’s the same side. 

Figure 1: Penalty kick

http://gifrific.com/wp-content/uploads/2012/06/Andrea-Pirlo-Chip-Shot-Penalty-Shot-Euro-2012-Vs-England.gif

For our purposes, we’ll assume the following:

1. To make things interesting, we’ll say the goalie is a guy and the striker is a girl.  This obviously isn’t true in a real game (unless you play co-ed pick-up ;p) but it makes it easier to distinguish who we are talking about.

2. The goalie dives either left or right (we’ll use the striker’s perspective throughout).

3. The striker shoots left or right.  The shooter won’t ever kick it in the center to the goalie…that’s a big no no.  We won’t worry about the case where she kicks the ball in the air or on the ground either.

4. If the striker kicks the ball to the same side the goalie dives, we’ll assume the shot is blocked.  If the striker kicks the ball to the opposite side the goalie dives, we’ll assume it went in the net and is a goal.

Pure or Mixed Strategy?

cThis setup is similar to the game of matching pennies.  In matching pennies, you and an opponent simultaneously reveal a penny.  If both pennies show heads, your opponent pays you $1.  If one penny is heads and the other is tails then you pay your opponent $1.  In this case, you can be viewed as the goalie while your opponent is the striker in our soccer example.

Figure 2: Payoff matrices for matching pennies and penalty kicks

https://williamspaniel.com/2014/06/12/the-game-theory-of-soccer-penalty-kicks/

Figure 2 shows simplified payoff matrices for matching pennies and penalty kicks.  As we can see, this is a zero-sum game because each of the four instances in the matrix adds up to 0.  It turns out that there aren’t going to be any pure strategy Nash equilibrium.  Why?  I’ll go over this in my presentation and we’ll look at each instance to answer this question.  Feel free to comment on why that might be!

We learned from Nevin’s presentation on Nash’s Theorem that there must be at least one Nash Equilibrium for all finite games.  Since there is no pure strategy Nash equilibrium then one must exist for mixed strategies which involve a probability distribution over two or more pure strategies.   This means that the players will randomly choose among their two choices (left or right).

So, what should the players do?  What might be an optimal strategy?  It turns out that both players should pick a side with equal probability.  This means the striker kicks left ½ the time and right ½ the time and the goalie dives left ½ the time and right ½ the time.  Why is this an optimal strategy?  It is because the players are less likely to figure out what their opponent will choose.  It allows the players to be less predictable.  

Figure 3: Payoff matrix with probability 50%

https://williamspaniel.com/2014/06/12/the-game-theory-of-soccer-penalty-kicks/

This is, in fact, an example of a Nash equilibrium because both players can’t gain by changing their strategy.  If the striker has a 50/50 chance of shooting the ball either to the left or right then it doesn’t matter which way the goalie dives because he will have an equal chance of stopping the ball.  Similarly, if the goalie is diving left or right with equal 50% probability then it doesn’t matter which side the striker chooses to shoot the ball because she will have a 50% chance of scoring.  By randomizing your options, you become less predictable to your opponent.

This is easier to see if we look at other strategies that are not optimal where probabilities are different throughout.  If the striker kicked left 100% of the time then it would be easy for the goalie to stop the shot because he would then dive left 100% of the time.  This is also true if the striker shoots left 99% or 98% and so forth.  The goalie will always want to go to the left.

Conclusion

In actuality, both players have a choice of more than two tactics (going left or right). The shooter might aim low or high.  She might shoot to the left, right or center.  She might also go for either power or accuracy (if you didn’t already know, it is very easy to boot the ball with too much power or aim too high which forces the ball to go waaay over the net.  Not good my friends!).  The optimal strategy for both players will be mixed but it is best that players keep their probabilities of choosing left or right as close to 50% as possible.

We will see that game theory can be applied to many other sports including baseball and tennis.  Can you think of any other strategic interactions that are found in other sports?

Sources

Harford, Tim. "World Cup Game Theory." What economics tells us about penalty kicks. June 24, 2006. Accessed April 12, 2017. http://www.slate.com/articles/arts/the_undercover_economist/2006/06/world_cup_game_theory.html.

Spaniel, William. Game Theory 101: Soccer Penalty Kicks. June 16, 2010. Accessed April 12, 2017. https://www.youtube.com/watch?v=OTs5JX6Tut4.

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