The Source

I’ve finally begun the process of revising and piecing together my portfolio, and as I do this, I’ve found myself reflecting on all of the things that got me here. So, for this post, I thought I’d give you all a little bit of insight into the source of it all: the art that has shaped my art from the very beginning. Somehow, this seems an appropriate thing to do as I approach the moment in my life when it will all culminate into a portfolio which may very well determine the course of my future.

So lets go back.

As a child, I clearly remember my five favorite stories being Harry Potter, Where the Sidewalk Ends, the Lord of the Rings, Cyrus the Unsinkable Sea Serpent, and Avatar the Last Airbender. 

The influence of Harry Potter and the Lord of the Rings (by J.K. Rowling and J.R.R. Tolkien, respectively) is perhaps the most self-explanatory. They both feature fantastical creatures and magical worlds, full of great beauty and grotesqueness. The Lord of the Rings is widely regarded as the birthplace of modern fantasy, because it was the first story to take all of the creatures and magic from fairy tales and put them together into one comprehensive tapestry of a story, shaping a whole world with its own languages, history, ethnography, and geography. It is easy to see the influence of Lord of the Rings in Harry Potter, and perhaps even easier to see their influence on me. One example I can give is that of Ungoliant, Shelob, and Aragog: the giant spiders. In the Silmarillon, Tolkien’s history of Middle Earth that sets the stage for Lord of the Rings, Ungoliant is the mother of darkness, a massive spider who lives to consume all within her reach. Later, in the Lord of the Rings, we are introduced to one of her distant offspring, Shelob, who, like her mother, is a large (though not nearly as big as her mother) spider who dwells in the darkness and has an insatiable appetite. In Harry Potter, Rowling takes another angle on Tolkien’s giant spiders, instead fashioning the character of Aragog into a wise and benevolent soul who lives peacefully in the forest. To see the influence of these creatures in my art one need look no further than my recently invented giant spider-slugs. In fact, when painting my first spider-slug painting, I used shots from the scene in the Lord of the Rings movie in which Sam and Frodo escape Shelob’s lair.

In addition to giant spiders, Harry Potter is known for a host of other bizarre creatures, including Hypogriffs, Thestrals, and House Elves, to name just a few. These too have added to my love of strange and unusual beasts. Here is a collection of all of Rowling’s creatures.

Moving on from Harry Potter, I come to Avatar the Last Airbender. This may seem silly, but hear me out. Avatar was my favorite TV show as a kid, and one of the quirky hallmarks of the world it took place in was that there were no ordinary animals. Armadillo Lions? Check. Badger Moles? Of course. But something normal? A bear, perhaps? No way. This splicing of known animals to form some oddly charming hybrid is something I have adopted, with my spider-slugs and snake-turtles.

A list of Avatar’s creatures can be found here.

To look, however, even further back, we come to Cyrus the Unsinkable Sea Serpent and Where the Sidewalk Ends. Both of these books were books my parents used to read to me as bedtime stories when I was very very young. Cyrus is perhaps the least creature-filled of all the stories I present here, but its impact remains undiminished. Along with Smaug (from Lord of the Rings), Cyrus was the first dragon (or dragon-like creature, because technically he’s a sea serpent) I ever encountered, and he certainly helped lay the foundations for my life-long dragon obsession. In that way, he will forever be with me.

Shel Silverstein’s Where the Sidewalk Ends, has no dragons, and not even many creatures, but certainly has its own flavor of the strange and unusual. From a two-faced man, to The Worst, to a dentist who finds pleasure in pulling out the teeth of a poor crocodile, to a man writing a poem from inside a lion, it has it all. As wonderful as the poetry itself is, the illustrations have always been my favorite part. And so this too from a young age shaped my affinity for macabre oddities.

 

There are certainly other stories and artists which have influenced me, the world itself is a constant influence, but none I think that have done quite as much as these four. I do hope I can do their legacies justice.

 

Image sources:

“Aiya Eärendil Elenion Ancalima!” Elvenesse. www.elvenesse.net/blog/tag/ents/.

Pete, Bill. “Cyrus the Unsinkable Sea Serpent” Pinterest. 

“Dragon Moose.” Avatar Wiki. http://avatar.wikia.com/wiki/Dragon_moose.

“Goat Gorilla.” Avatar Wiki. http://avatar.wikia.com/wiki/Goat_gorilla.

“Giant Rhinoceros Beetle.” Avatar Wiki. http://avatar.wikia.com/wiki/Giant_rhinoceros_beetle.

Silverstein, Shel. “The Worst.” Pinterest.

Silverstein, Shel. “It’s Dark in Here.” Pinterest.

The Theater of it All

As I stepped into the studio yesterday, I knew that I wasn’t feeling it. My body and moving were not things that I wanted paired. But I had to get going. So I thought, “How do I loosen myself up?” What I ended up doing was putting on some Frank Ocean and dancing my ever-loving heart out. I didn’t think about my movements (or anything else for that matter.) I just moved. I danced and danced as song after song played. While a lot of the movements repeated, and I tended to burn out 3/4 of the way through the song, I still managed to develop quite a few good movements. The song Provider is what really got me moving. Something about the line “show me the wisdom in your movement” really kept me inspired. I came in sluggish and by the end all I wanted to do was dance.

wall_decal_theater_masks_s

Image from: http://2017.satf.us/

This exercise really got me thinking about the theater of dancing. When I was anticipating each lyric, and moving my body in a way that could best represent each word, phrase, and sound, I was able to realize just how similar this is to theater. I’m trying to convey emotions set out for me in my own unique way. My perspective is the guide, but it is heavily influenced by the work of others (or in the case of my dance, myself.) I remember, a few weeks ago, I was talking with T. Will about my project, and he said to me jokingly “You couldn’t incorporate theater into your project?” But the more I think about it, dance is theater. Dance is just a different manifestation of the same idea. That knowledge helped guide me through choreography. I kept thinking to myself, “how would I act this out?” And I let the movement take me where it needed to go.

How on Earth

This past week I ran into a pretty big issue that may take a little time to resolve, it’s honestly pretty embarrassing to me but I’ll just go ahead and put it out there: I don’t remember how to paint portraits. Imprimaturas? Fine and good. But actually putting color on the canvas to resemble skin tone, depth, shape, and shadow? I’m having a harder time getting back into it that I thought I would. When I actually think about it, it’s been probably about two years since I last painted a portrait. Here’s the comparison between my two year old portrait (still unfinished) and my progress on the Maggie one:

It’ll come along, I just need to take some uninterrupted time to sit and work on it. I’m thinking of playing up some of the yellow highlights, like it appears in my reference photo. I do like how the eye is coming, though.

This past weekend, I began the process of migrating all of my portfolio works over from home to Westtown, so that I can begin photographing them properly to upload into my college applications. I have an email into T. Chris Willis to set up a time to meet about beginning this photographing process. I also need to contact T. Joyce Nagata about photographing my pottery. I’ve found two good articles on photographing paintings here and here. It looks like I may also need to contact T. Sarah Sullivan about setting up some lights with gel filters to balance color.

In looking ahead, I have some ideas for new paintings. I know I don’t want to stick solely to portraiture and abstract and fantasy separately, but rather I would like to merge the three to start to string together a coherent body of work. I am still mulling over how I will do this with the Maggie painting, but I am thinking of pulling in some elements of my floating amoebas with abstract tendrils of paint and a focus on mixing color.

Next, I am planning on doing a self portrait with poison ivy and a spider-slug. I know, I know. You’re wondering about the poison ivy. Well, in addition to gathering up portfolio pieces over the weekend, I also waged an itchy and exhausting battle against the poison ivy covering my arms, torso, and yes, even my face. I did, however, get some pretty gnarly shots of my eye all swelled up, and I thought it might make for some interesting and out of the ordinary subject matter (like the grotesque work of Jenny Saville which I mentioned in an earlier post). I also am looking to start bringing my creatures into realism, and what better way to do so than with a spider-slug on my head?

I am toying with the notion of doing a series of portraits of my friends, all playing with the abstract and fantastical. I think I will consult T. Chris Willis on this, but I am feeling hopeful. This weekend I will be staying at school and am planning on holing myself up in the studio to get some actual work done. Hopefully, I will be able to finish or get closer to finished on the Maggie painting, and begin the self-portrait.

Assistant Teaching – Alec Barbera

My time in the 6th grade classroom is quickly becoming a part of me in that I am connecting with the kids much more now. I feel like they trust me now and they are much more comfortable with asking me questions and having me help them. I feel more and more that this is no longer just a project, but something that makes me get up in the morning and smile while walking through the entrance of the middle school.

Continue reading

Richard II’s Policy in the Localities, 1389-1397 -Gwyneth

For this week’s blog post, I want to get back to talking specifically about the work I have been doing on my independent project. So far, this has consisted primarily of reading. Last week, I finished a particularly interesting chapter in my current text, a local study of the West Midlands region by A.K Gundy entitled Richard II and the Rebel Earl, about Richard II’s attempts to increase royal control of the localities from 1389-1397. Typically, Richard’s policy of building up local authority is associated only with the period referred to as his “tyranny”, which lasted from 1397 until his deposition in 1399. However, Gundy makes the argument that Richard had been pursuing this policy since 1389, when he first regained power after being placed under significant restraints following the Lords Appellants’ coup in 1388. Overall, I think Gundy’s argument is very compelling and may have significant implications for the way that Richard’s transition into tyranny is viewed.

To begin my explanation of Gundy’s argument, it is necessary that I provide a bit of context about the political environment of the late 14th century. Although it is often assumed that medieval kings exercised absolute authority over their subjects, this was far from the case. In fact, the monarch depended on the support of the elite noble and gentry classes, whose “land gave them the ability to raise an armed force from their tenants to enforce the king’s rule” (Gundy 19). Furthermore, the existence of parliament, which had the power to grant (or to not grant) the taxes necessary to fund royal policies, “meant that [medieval] government was necessarily consensual, needing the support of an articulate and informed propertied class” (Harriss 39). Thus, Richard II did not possess absolute power, but was held accountable to the nobility and gentry who ruled the localities. Paranoid by the ease with which his nobles had stripped away his power during the crisis of 1388, Richard sought to subvert this check on his authority by replacing the influence of the nobility in the localities with his own influence.

 

 

historic-counties-map

Map showing the historic counties of Great Britain (Worcestershire and Warwickshire are located towards the south-west)

Gundy argues that Richard’s policy in the 1390s sought to erode the power of his opponents by replacing their affinities as sources for local officers, as well as discrediting them publically (Gundy 164). One tactic he used in pursuit of these goals was controlling the appointment of local officials (Gundy 159), two of the most important of which were the sheriff and the Justices of the Peace, or JPs (see here for more specific information about local offices). In the two counties examined in Gundy’s study, Warwickshire and Worcestershire, Richard aimed to limit the positions held by men loyal to the dominant magnate, the Earl of Warwick, who had been one of the leaders of the 1388 coup. For example, of the eight men appointed to the shrievalty of Warwickshire (which was held jointly with neighboring Leicestershire) between 1389 and 1397, none came from Warwick’s affinity. Furthermore, one of these sheriffs was connected with Richard himself, a second with the king’s ally Thomas Mowbray, while a third and possibly a fourth had ties to Richard’s uncle, the Duke of Lancaster (Gundy 161-162). In Worcestershire, Richard was met with a greater challenge, as Warwick’s influence was more absolute than in Warwickshire and he held the position of sheriff permanently, since it was passed down hereditarily in this particular county. Richard targeted Warwick’s influence through the peace commission, which was restructured in 1393 with fewer men from Warwick’s affinity and in their place, more men close to the king (Gundy 173). Warwick’s response to these appointments lends validity to Gundy’s argument that these moves were deliberate attempts by Richard to push Warwick’s men out of local office. Breaking with the typical behavior of a hereditary sheriff, in the 1390s Warwick began presiding as sheriff of Worcestershire in person rather than using a deputy for the first rather (Gundy 176-177). Similarly, in Warwickshire, where Warwick was a frequent JP, the 1390s saw him acting as JP in person more frequently than in the period from 1374-89 (Gundy 181). The implication of Warwick’s increased use of his local offices during the 1390s is that he was attempting to use his own authority to check the clear intrusion of royal influence into his localities.

According to Gundy, Richard made significant use of two more tactics in his attempt to increase his local power during the 1390s: “manipulation of legal procedure and supporting his own men in local disputes” (Gundy 159-160). Since legal procedure was the typical means through which local disputes were resolved, these two strategies are closely intertwined and thus can be examined together. One way in which Richard used legal procedure to his advantage was by prosecuting his opponents though means outside the common law, such as his council. This allowed him to take opponents by surprise and leave them unprepared to defend themselves (164). Richard also attempted to further his interests in the localities by addressing disturbances with commissions specifically designed to bring about a particular outcome. For example, Richard excluded Warwick’s men and instead appointed “an increasing number of outsiders” to various commissions issued in the 1390s pertaining to a dispute between royal officials and the bishop of Worcester over land in Worcestershire (Gundy 163). Additionally, a commission appointed to investigate the misdeeds of one of Richard’s most trusted retainers, William Bagot, was purposefully given limited powers that went so far as to prevent it from inflicting any punishment upon Bagot (Gundy 184).

After reading this chapter, I feel that Gundy’s argument has a good deal of potential. It offers a compelling explanation for Richard’s sudden slip into tyranny in 1397 – that it was not sudden at all, but rather the final stage of a policy he had been unrolling for years. However, before I can come to a conclusion regarding the validity of this argument, I feel that I need to look further into the specifics of royal intervention in the localities from earlier in the reign and from the reigns of different monarchs in order to deduce just how unusual Richard’s tactics were.

 

Sources

Gundy, A. K. Richard II and the Rebel Earl. Cambridge UP, 2013. Cambridge
Studies in Medieval Life and Thought, Fourth Series.

Harriss, Gerald. “Political Society and the Growth of Government in Late Medieval   England.” Past & Present, no. 138, 1993, pp. 28–57. JSTOR, JSTOR, http://www.jstor.org/stable/651187.

 

Image

“Historic Counties of Great Britain.” Pictures of England, 2017,
http://www.picturesofengland.com/mapofengland/historic-uk-counties.html. Accessed
17 Oct. 2017.

 

 

Moving below the surface (2): Cross-entropy — William

In my previous post, cross-entropy is presented as a cost function, measuring the difference between given inputs and outputs in supervised learning. It is also an important concept from the Information Theory.

Before we talk about what cross-entropy is, let me introduce John, an imaginary freshman entering Westtown. He loves desserts and talks a lot about it. In fact, all he says are none but four words: ice-cream, chocolate, cookie, and yogurt. When he is back home after school, however, John only use binary code to text with his classmates. So, the texts his classmates received looks like this:

binary_bits

To understand what the messages mean, John and his classmates need an established code system, a way to map sequences of binary bits into words. Here is a simple example:

Code_mapping

With the code system in hand, John and his classmates can encode text messages by simply substituting words for codewords and vice versa.

encoding

It turns out that John does not use all of his four words as often. As an ice-cream fan, John shares his ice-cream-loving moments all the time. He sometimes mentions his time eating chocolate ice-cream with his family, and rarely does he mention anything else. So, his word frequency chart looks like the following:

milton_word.png

Now we can plot out John’s word frequency against the number of binary bits each symbol matches. The area formed represents John’s average length of messages we use to send each word. These areas are formally named entropy.

milton_code.png

Since typing and sending text message takes up time for John and his classmates, they want to reduce average codeword length for each word so they could spend minimum time texting the same amount of words. This is when variable-length code comes into play, mapping commonly used words (like “ice-cream”) to shorter codewords and less-frequently used ones to longer codewords (like “cookie” or “yogurt”). So, we have a new mapping between words and codewords:

new_code.png

Side note: the mapping is not randomly picked — it is designed as a function of the word frequency so it is uniquely decodable and not cause confusion when splitting the message into codewords.

Again, we could plot the word frequency against the number of binary bits. Here is how it looks like:

new_code.png

The result is amazing: we successfully reduce the average length to 1.75 bits! It will be the most optimized code in this case. Encoding words with it will take up a minimum number of bits and the text messages will be the most concise.

Very soon after the school started, John meets Juliet, another freshman at Westtown. Juliet is a chocolate-lover. She talks about chocolate all day, and her favourite is chocolate chip cookies. Juliet despises ice-cream though, and mentions it only when necessary. Despite this, they share their obsession with dessert and, interestingly, the same limited vocabulary size.

Juliet_word.png

When Juliet started to use John’s code, unfortunately, the text messages she sends are much longer than John’s, since they have different word frequencies. As we plot Juliet’s entropy graph, we could see her average message length is as long as 2.25 bits! We call Juliet’s average message length using John’s code system the cross-entropy.

So, why do we care about cross-entropy in supervised machine learning? Well, cross-entropy provides us a way to measure the difference between the result our model produces and the provided outcome. Since both the result and the provided outcome could be both expressed in the form of frequency charts or possibility chars, cross-entropy fits its role as the cost function well. With the cost calculated, we could adjust the parameters to achieve a best-fit model for the given outcomes.

See you next week!

Work Cited

“I like this Maple Application – Vibration of Mindlin rectangular plates.” Vibration of Mindlin rectangular plates – Application Center, http://www.maplesoft.com/applications/view.aspx?SID=35302&view=html.

What is bit (binary digit)? – Definition from WhatIs.com. (n.d.). Retrieved October 14, 2017, from http://whatis.techtarget.com/definition/bit-binary-digit

Formal Ink Painting and Folk Painting

As I mentioned in last week’s blog, this week, I explored the difference between Korean folk paintings and Korean formal and orthodox ink paintings that include Sumukhwa, which is the painting in ink monochrome.

From the book, “Handbook of Korean Art,” I read the chapter called “Appreciating Folk Paintings.” It underlines how invaluable Korean folk paintings are since they “portray the simple and unaffected daily lives of ordinary people” that include religion, mythology, and the mindset of Koreans. This line reminded me of the conversation that I had with Teacher Joyce. We discussed different techniques in drawing ink paintings and folk paintings. As ink paintings are more sophisticated, drawn by people of high social status during the 17th Century, they require more education and practices to paint. If I were to paint ink paintings like Sumukhwa, which were created for purely artistic appreciation, I would have had to start formally drawing since I was young. So, adults and I who try to draw Korean paintings as a hobby find it difficult to draw formal ink paintings.

Meanwhile, Korean folk paintings allow anyone to enjoy drawing them, as the common people did during the 17th Century. Also, they are “informal artwork that represent people’s thoughts and emotions.” So, the audience can understand more deeply and clearly about how Koreans lived in the past.

I will show you some examples of formal ink paintings that are painted in Korea (on the top), China (on the bottom left), and Japan (on the bottom right) under each theme: cherry blossom and tigers. (Each painting’s countries are distinguishable by written characters as well)

However, interestingly, under the same theme, each painting from three different countries looks similar. It is because as three countries, Korea, China, and Japan, are geologically close to each other, there has been frequent contact and trading since the far past. So, they share similar ideologies, religions, and traditions such as temples, Buddhism, and even arts. Korean formal painting Sumukhwa was influenced a lot by Chinese artistic habits. In the end, the three countries’ ink paintings apparently look similar.

But there is something significant about Korean folk paintings that differentiates them from the two other countries. As I have shown you with the Korean folk painting Hojakdo that portrays tiger from week 2 blog, it is more informal than the formal ink painting of tiger but still has its unique characteristic. Unfortunately, as the book “Handbook of Korean Art” indicates, while Korean folk paintings are recognized as a genre of painting, “they have never been fully studied in art history.” It is because most of the artists and dates are not known and viewed not as historical records in Korea. It is however welcoming news that recently, Koreans including myself have started to show increasing interest in Korean folk paintings. I hope formal research on Korean folk paintings is more widely done and that this fascinating art becomes more recognized and popular.

Next week, I am going to explore the development and influence of Minhwa in modern Korean society.

Thank you for reading my blog!

Citation:

Yoon, Yeol-su, et al. Handbook of Korean Art. Yekyong Publishing, 2002.

“임진성 수묵화 (Imjinsung Sumukhwa).” 블로그 홈 (Blog Home), blog.daum.net/_blog/ArticleCateList.do?blogid=04Lo9&CATEGORYID=748397.

Liu, Lian. “Crafts for Kids.” Pinterest, 12 Nov. 2014, http://www.pinterest.com/pin/299982025153884546/.

“Japanese.” Pinterest, http://www.pinterest.com/teresaenlasnubs/japanese/.

“Draw Tiger by 8 Big Painter of Japan and China Book Ink Wash Painting.” Books WASABI, http://www.books-wasabi.com/product/829.

http://www.cybermusee.com/case/cybergallery/blog1.asp?mh=choi1308&mh_hname=Choi%20Kyonglak

Drew, Vince. “Art I Love.” Pinterest, 3 Sept. 2014, http://www.pinterest.com/pin/436778863834435769/.

 

The Importance of Press – KC

On October 2nd 2017 a Q&A about the work I’ve been doing was posed on Vice News. It was later added to their national snapchat story. It’s hard to say how many people saw the article but this was national coverage which means a TON of people saw it all across the country and perhaps the world.

Let’s look at the numbers we do have:

We can use Facebook’s article tracking feature to see how many times it was simply shared on the popular social media site. In the past two weeks it has been shared by nearly 2,000 people and popular pages.

Screenshot at Oct 15 19-54-20.png

I don’t have any way to quantify any other post-based social media websites like twitter, but this gives an audience rage on one.

We can however extract a few numbers from Snapchat’s stories. Vice News is one of the most popular snapchat stories, and while the app does not release official viewership counts, NBC released their own count earlier this year.

Read more: Quakerism’s Influence on my Activism – KC

According to Variety, the multi-media giant garnered a whopping 29 million views in the first month of starting their new snapchat story. While this number is probably inflated because of first month promotion, it allows us to see the amount of people who are tuning into a specific story – a new one at that.

It is safe to say that over a hundred thousand people saw the story on Vice. We don’t have any way of quantifying the number of people who then chose to read the article, but they were all able to see this video:

 

So why do these numbers matter? It’s simple, good press is one of the most crucial parts of any organization or movement. Over the past two weeks since my article dropped, my mailbox has been flooded with new people wanting to get involved. Leading activist in my field have begun reaching out to partner.

Read more: Timing is an Art Form – KC

I’m really excited about working with these people and continuing to build my organization. To those trying to build something new, I suggest you start working on news coverage. Reach out to local reporters or people who frequently write about related topics. Start sending press-releases when new things happen inside your organization.

These kinds of articles will help propel your message and build a wider audience.

Groundbreaking of the mushroom production facility in China

I had some difficulty getting through my project in the past two weeks since the Commissioner was out in China for the groundbreaking of the mushroom production facility. However, I am able to track his visit and proceed with my project from here. According to my recent communication with him, they had a successful launching activity in China and the activity serves as a starting place for future cooperation opportunities.

Funan is where the groundbreaking event took place. The county is located in Anhui Province. Its unique location and abundant resources allow Funan to become an ideal place for producing mushrooms. At present, the county has a cultivated fungus area of 1.2 million acres. The area consists of 7 different kinds of mushrooms, of which four products are recognized as pollution-free product, and the other three are given green product certificates. Funan Lianmei company is an American company based in Funan for the production of mushrooms. The company is planning on investing 1 billion Yuan and use advanced technology in America to achieve an annual production of 30,000 tons of mushrooms. If all goes well, they will be able to gain an annual profit of 100 million Yuan, and provide 1000 job opportunities for the local community. To read more about the cooperation, click here(you might need some translating tools, I was not able to find the same article in English). The picture below captures the groundbreaking event.

03.JPG

It is interesting to see how a small county in China gets into business partnership with a company in Chester County area. I think China Initiative is definitely powerful and will bring more prosperity for both China and America. I will be blogging more about this cooperation after talking about it more with Commissioner Farrell.

In the meantime, I had a conference with his assistant last week about some plans moving forward.Commissioner Terrence Farrell has gained permission from the other two commissioners to establish a Sister Municipality agreement. Their decision allows me to move forward as a liaison between both places. I am very excited for the future development of this project.

Picture: http://kfq.ahfn.gov.cn/content/detail/59dd67727f8b9a6e6a060091.html

 

Game Theory in Tennis–Summer

maxresdefault.jpg

Today, I’m going to explore with you, through the lens of Game Theories, strategies within tennis.

As we know, in a tennis match, players are trying to score points against each other by hitting the ball into a corner of the opponent’s court where s/he is not able to hit it back.

Let’s focus our attention on a pass between two tennis players. The server, player 1,  has two choices: passing towards the receiver’s left (backhand side) or right (forehand side, assuming right-handed). The receiver at net, player 2, also has two choices: leaning left or right to catch the ball. Because each player is differently skilled at backhand and forehand and player’s satisfaction depends on his possibility of scoring/preventing his opponent from scoring, we have the following payoff matrix:

Player 1(left)/2 (top) Left Right
Left 50,50 80,20
Right 90,10 20,80

In this matrix, we cannot see any strictly/weakly dominated strategy or traditional Nash Equilibrium. In fact, this observation coincides with our real world experience: a tennis match is never good if the players always lean to one specific side!

In this tennis game, however, Nash Equilibrium still exists, but in such a Nash Equilibrium both players are playing, what game theory calls, Mixed Strategies. A mixed strategy is a randomization of available strategies that assign each strategy (left/right) a possibility p such that 0‹p‹1 and all p add up to 1. For example, a possible mixed strategy for player 1 is hitting left 20% of the time and hitting right 80% of the time.

The payoff of a mixed strategy is logically given by the sum of the probabilities of individual strategies time their respective expected payoffs against the opponent’s strategy. For instance, if player 1 plays the mixed strategy (0.2, 0.8) and player 2 plays the mixed strategy (0.7,0.3), then the payoff of player 1’s mixed strategy is:

0.2×(0.7×50+0.3×80)+0.8×(0.7×90+0.3×20)=67

So, given all these definitions, how do we find the mixed-strategy equilibrium of this game? We have to infer a step further.

If the two players are at Nash Equilibrium, then both of them are playing their best responses to the other’s strategy. If a mixed strategy is a best response, then individual strategies within the mix must also all be best responses, or else we can create a better response by taking the strategy with a lower payoff out of the mix. Thus, the only way both hitting left and right are best responses is that their expected payoff  are equal.

Let’s notate player 1’s equilibrium mixed strategy as (p, 1-p) and player 2’s equilibrium mixed strategy as (q, 1-q). Given our inference, if we want to calculate player 1’s equilibrium mix, we can look at player 2’s expected payoffs. Because player 2 is playing a mixed-strategy best response, then as we previously concluded, his expected payoff of leaning right and left should be equal. Thus, we have:

Left   50p+10(1-p)=20p+80(1-p)   Right  

We can solve this equation: p=0.7, 1-p=0.3. Thus, player 1’s mix strategy at Nash Equilibrium is (0.7,0.3).

Similarly, we can solve for player 2’s Nash Equilibrium mixed strategy with player 1’s payoff.

50q+80(1-q)=90q+20(1-q)      q= 0.6, 1-q=0.4

Thus, the game has a mixed strategy Nash Equilibrium at {(0.7,0.3), (0.6,0.4)}.

This model draws heavily on the concepts I’ve introduced in my previous posts and has slightly more calculations and variables than my previous examples. Yet, despite its being slightly more complex, it is by far the most accurate representation of the choices we face in sports and in life.

See you next week!

Work Cited: Basic rules of tennis. (n.d.). Retrieved October 14, 2017, from https://www.myactivesg.com/sports/tennis/how-to-play/tennis-rules/basic-rules-of-tennis
Tennis Groundstrokes – The Forehand and Backhand Groundstroke. (2015, October 30). Retrieved October 14, 2017, from http://www.optimumtennis.net/tennis-groundstrokes.htm
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