Christopher X J. Jensen
Associate Professor, Pratt Institute

Evolutionary Games Infographic Project: New “conceptual” images

Posted 06 Feb 2012 / 0
UPDATE: The images discussed below are now available for free use on the Evolutionary Games Infographic Project page.

For the past two semesters I have been working with Greg Riestenberg, a graduate student in Pratt’s Communications Design program, to come up with a new series of images designed to explain some common evolutionary games (I am calling this the “Evolutionary Games Infographics” project). Greg is providing most of the design ideas and all of the design expertise; I am providing the perspective of our target user, a teacher (or self-directed learner) who needs better images to explain game theory. Our eventual goal is to place these images on Wikimedia Commons so that anyone can use them in lessons, presentations, and even publications (for free!). We are still exploring our options on Creative Commons, but chances are we will use the CC BY-SA license, which would require that any work containing our images be attributed and only modified if they are shared in a similarly open manner. More on this once we finalize a set of images. The purpose of this post is to provide a “preview” of these images. Obviously (given that these are on the web) we welcome anyone interested to view and comment on these images. However, we are particularly interested in feedback from our target audience: teachers and writers who need intuitive, easy-to-understand conceptual graphics describing classic evolutionary games. These graphics are designed to be “modular” so that they can be used in a variety of contexts and a variety of visual configurations. If you can take the time to review the images below (including my narrative explaining what we sought to accomplish with each image), we would greatly appreciate your feedback. Feel free to use the “comment” function at the end of this post, to email us directly, or to use my contact form. As I have discussed in a previous post, the existing graphics used to depict evolutionary games suffer from a number of problems. The first problem is that they are often poorly crafted and visually unappealing, containing distracting elements that serve little or no communicative purpose. Many of these graphics contain a combination of images and numbers, which begs the question: why not just use numbers? In fact, most books and webpages rely on the typical communication device of game theory: a matrix representing the numerical payoffs of different combinations of ‘choices’ in the game. This is fine for those who instantly gravitate towards numerical representations of the world, but what exists for the rest of us? Greg and I set out to create a series of images that use only visual elements to communicate the key outcomes of three historically-important evolutionary games: the prisoner’s dilemma game (PD), the hawk-dove game (HD), and the stag hunt game (SH). We wanted to create images that could be used in a variety of contexts to effectively communicate how these games work, what dynamics emerge from the game, and how the possible outcomes of each game differ. The goal was not to create ‘stand alone’ graphics that fully explain the game, but to provide graphical aids that — when paired with appropriate spoken or written words — make these evolutionary games more intuitive.

We have made a series of eight images representing each of the evolutionary games. To show how each of these images works, I will provide narrative on the critical features of each image, using our PD set as an example. All of these images share a common visual key, which can accompany any of the images:

The first image we provide for each game is a conceptualized version of the classic matrix representation of two-player games:

This image uses color hue, color intensity, shape, line characteristics, and fractional shading to allow the viewer to better understand how the PD works. Players are differentiated in two ways: each has a unique color and unique shape. In our representation of the game, payoff or ‘score’ is depicted by varying degrees of white fill in each player’s shape: the more filled with white you are, the better your outcome. In the PD, the best possible outcome for each individual player is to defect when her partner cooperates. In the the top right and bottom left panels representing this possible outcome, the color of the defecting player is shown as intensified, signalling the improvement of that player’s condition. In addition, the defecting player also earns a completely-filled shape, which contrasts strongly with the ’empty’ shape of the other player. The dashed line of the defecting player contrasts with the solid line of the cooperating player, further highlighting the different outcomes enjoyed by each strategy. The second-best individual outcome in the PD is for each player to cooperate; this is also the best overall outcome for any pair of players. We represent this shared outcome by showing each player as 75% filled and by using a high-intensity color that is the blend of the hues representing each player. The second-worst individual outcome is also a shared outcome, one produced when each player defects. We depict this poorer shared outcome by showing each player as 25% filled in a low-intensity background that is also a blend of the hues representing each player. To make it easier to identify which game this matrix corresponds to, we have created a little “PD” icon in the top left corner. Each game has its own ‘brand icon’, which allows for easy side-by-side comparison of the games.

While I have decomposed the entire visual strategy behind these images in the rather long paragraph above, I want to emphasize that it is not necessary for learners to explicitly see these elements in order to learn from these images. Our brains instinctively look for patterns, often recognizing them without being conscious of that recognition, and so our goal was to provide the simplest patterns that make each of these games as intuitive as possible. As the images for the SH and HD below will make clear, we have tried to retain a common ‘visual vocabulary’ throughout this series of images, changing patterns only where the outcomes of the games vary.

The second in our series of images for each evolutionary game allows learners to compare the overall quality of individual outcomes from the perspective of each of the two players:

The hues of each player are again used to show how outcomes vary for each player, and the ‘brand’ icon shows that this is a representation of the PD. Because we used dashed lines to represent defection and solid lines to represent cooperation, these outcomes can be pulled out of the traditional matrix and rearranged without losing their meaning; the remaining images take advantage of this flexibility. The introduction of greater-than signs ranks outcomes for each player. A critical characteristic of the PD is clearly portrayed in this image, as the best outcome from the perspective of one player is the worst outcome from the perspective of the other. The two egalitarian outcomes — both players cooperating or both players defecting — are shown in parallel as the second- and third-ranked outcomes.

Because a major focus of these evolutionary games is to compare how individual and collective motivations can be in opposition, we also show a ‘ranked’ image representing the best-to-worst overall outcomes as measured by the total payoff for each pair of players:

That the four possible overall outcomes of these games are not equivalent is key: this image allows teachers to show that these evolutionary games are non-zero-sum. This image would also be particularly important in explaining the results of iterated games, which are often dictated by which combination of player moves results in the best overall outcome.

We expect these three images above to be the most useful, as they lay out the dynamics of each of the evolutionary games. But some teachers may wish to also use these visual depictions of the game to bridge to the more traditional mathematical representations of these games. To allow this kind of bridging, we also provide an image representing the ‘extensive form’ of each game:

As with the traditional extensive form representation of these games, we use a dotted line to show that the blue player makes her choice without knowledge of the green player’s choice.

Frequently the payoffs of the PD are represented by four letters: T for temptation, R for reward, P for punishment, and S for sucker. To help teachers bridge to this representation of the PD, we provide an image that ties our iconic representations of game outcomes to these letters, again emphasizing the relative ordering of individual outcomes:

This image showing the payoff to each player can then be used to return to the normative (matrix) form, this time showing the letters representing each payoff:

As with the first image showing a normative form representation of the game, we retain color-coding as a means of differentiating player outcomes.

Some teachers or presenters may want to transition smoothly from our symbolic representation of evolutionary games to the more tradition numeric representations. We provide two images that allow this transition to be made. The first uses the stereotypical PD payoffs (T=5, R=3, P=1, S=0) in a completed matrix:

For those who wish to include their own payoff schemes, or who wish to compare variable payoff schemes, we also provide a blank matrix into which numbers can be inserted:

We envision users of these images having many options. They can be used in series (most likely in the order depicted above, but there may be a reason for them to be used in other orders) or ‘a la carte’ as deemed necessary.

A key feature of these images is that they allow learners to compare the dynamics and outcomes of these three important evolutionary games. The common visual vocabulary used in each set of images makes this sort of comparison easier. Below is a side-by-side comparison of the PD, HD, and SH in terms of their payoffs to each player:        Placing these images next to each other allows key differences between them to emerge visually. For instance, it is easy to see that the key difference between the PD and HD centers on whether the worst outcome is caused by being defected on while cooperating (as in the PD) or by mutual defection (as in the HD); looking to see where the 0% filled symbols reside quickly illuminates this difference. Similarly, the key contrast between the PD/HD and the SH can be seen by looking to see where the 100% filled symbols reside: for the PD/HD the best payoff is earned by defecting on a cooperating partner, whereas in the SH the best payoff is earned by mutual cooperation. Although these three evolutionary games often use different words for the two available strategies, we chose to retain the words “cooperate” and “defect” because in all three games there is a choice that benefits the pair (cooperation) and a choice with the potential to advantage one player over the other (defection). Another way of visualizing the key differences between these games is to place the ranked individual outcomes side-by-side:

Viewed in this manner, the variation between games in what is the worst outcome and what is the best outcome can be quickly detected as a shift in where particular color-coded outcomes are located. That cooperation is the best absolute individual outcome in the SH can be easily seen in contrast to the PD/HD, where individual defection on a cooperating partner is always best. Contrasts in the overall outcome for the pair are also easy to visualize by placing these images side-by-side:

Thus far, these are the images that we have created. We would appreciate any feedback on the efficacy, accuracy, and value of these images. Any suggestions for additional images would also be appreciated. We are currently developing images that explore paradigmatic ‘real-life’ examples of these games (for instance, an ‘escalating cold war between nations’ image for both the PD and the HD). We are also looking to create images for other influential evolutionary games. Please keep in mind that these images are not yet available for public use. As such we retain the copyright to all images. Please be patient while we complete our refinement process: as soon as we feel these images are complete, they will be made available for public use.

You can browse the entire Prisoner’s Dilemma (PD) set in the gallery below:

You can browse the entire Hawk-Dove game (HD) set in the gallery below: 

You can browse the entire Stag Hunt game (SH) set in the gallery below:

You can check out earlier versions of these images here.

Evolutionary Games Infographics, Game Theory, Information Design

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