UPDATE: The images discussed below are now available for free use on the Evolutionary Games Infographic Project page.
This semester 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. Our first images for the Evolutionary Games Infographic Project, posted here and here, were designed to make the differing payoffs of the Prisoner’s Dilemma (PD), Hawk-Dove (HD), and Stag Hunt (SH) games easier to understand. These images were predominantly based on conventional means of explaining these games: most of our images either represent or are derived from the ‘normal’ and ‘extensive’ form representations. While these representations are useful for understanding the various possible outcomes of these games, they are less valuable for understanding the dynamics of how the game is actually played. Understanding how the game is played becomes more important as we compare games with very different play sequences. For example, the PD, HD, and SH games all share a common sequence: players must simultaneously choose between two play options, and neither player has any chance to preview what the other player will choose (for this reason we can say that these are ‘symmetrical games’). In contrast, a game like the Ultimatum Game (UG) asks players to make non-simultaneous choices, and the nature of the choice afforded to each player differs (for this reason we can say that this is an ‘asymmetrical game’). We wanted to create graphics that would communicate how the sequences of these games differ. Below are the first ‘sequence’ images we have produced, for the PD, HD, SH, and UG. This image represents the sequence of play for the Prisoner’s Dilemma:
This image, and the analogous images below representing the HD and SH, use basic symbols to represent the stages of the game. We show these stages at the bottom of the image as a line with three discrete events on it: the condition before the game is played, the choice made by each player, and the result experienced by each player. These different events of the game are represented as different shapes. To assure that the important choice that drives the game is differentiated from the conditions before or the final result, we have represented the choice as an unfilled shape. Players are represented using the green and blue colors first introduced in our conceptual images, and the condition of each player is represented using simple facial expressions. Colored lines represent the trajectories followed by each player, and after the choice is made we represent cooperation with a solid line and defection with a dashed line. Players make a simultaneous choice, represented by the single large square, and then the result experienced by each player flows from the combination of their choice and the choice of the other player. We represent the four possible combinations as four sets of parallel lines emerging from the choice like separate train lines on a metro map. The distance traveled by each player along this line represents their resulting payoff. We provide a payoff scale at the bottom of the image; this payoff scale shows the relative ranking of different payoffs, and corresponds directly to the normal-form matrix showing these payoffs. To reinforce the relative value of the payoff, we also represent the player ‘reactions’ to their result with a variety of positive and negative facial expressions. This image represents the sequence of play for the Hawk-Dove game:
The major difference between the HD and the PD relates to the relative payoff for cooperating when one’s partner defects as compared to the payoff for mutual defection. In the HD mutual defection provides the lowest payoff, which is represented by the value “X” depicted on the payoff scale. This image represents the sequence of play for the Stag Hunt game:
The SH differs from the PD and HD in the payoff awarded for defecting when your partner cooperates relative to the payoff for mutual cooperation. Mutual cooperation leads to the highest payoff in the SH, so the image shows the two cooperators traveling the furthest distance along the payoff scale. A valuable feature of these images is that they can be lined up next to each other to demonstrate the similarities and differences between the games:
By placing the PD in the middle of these three stacked images, we can see that the games are slight variations of each other. These images could also be placed sequentially in a presentation, and scrolling through them reveals their overall similarities and key differences. In addition to being able to compare the subtle differences in payoffs that distinguish the PD, HD, and SH, we would also like to be able to compare them with other games with very different structures.
One such game is the Ultimatum Game. Portraying the UG by an analogous diagram presents some challenges. The first challenge is that players of the UG make their choices at separate junctures of the game, so the diagram is required to show the sequential timing of these choices. The second (and larger) challenge is that unlike the PD, HD, and SH, the UG does not involve a discrete set of player strategies. Strictly speaking, each player can employ an infinite set of strategies. The first player must choose how to divide up a resource, with possible strategies varying from keeping 100% of the resource (which means offering 0% to the other player) all the way to keeping 0% of the resource (which means offering 100% to the other player); any incremental offer within this range can be chosen. The second player must decide whether to accept the offer or to refuse the offer (in which case both players receive nothing). This might seem like a binary choice, analogous to the ‘cooperate or defect’ choice available in the PD, HD, and SH. But this is not a binary choice: what the second player is in fact doing is choosing from a set of infinitely-gradated choices, and this set is analogous to the choice set available to the player making the initial offer. The major difference for the second player comes not in the available range of choices, but in the meaning of that player’s choice: rather than being an offer, the second player’s choice is instead a threshold. If the second player chooses the threshold of 50%/50%, he is saying that he will refuse any offer in which he receives less than 50% of the available resource. We call this threshold a norm because it is an internalized valuation of what constitutes an acceptable offer. That the UG involves an infinite range of fine-gradated strategies employed by each player is problematic for our diagrams. Unlike the PD, HD, and SH — which each only allow four combinations of possible play — the UG has infinite possible outcomes. We obviously cannot represent infinite choices and the resulting infinite combination of outcomes, so we had to make careful decisions about how to show representative choices rather than the full array of possible choices. We did this by showing several offers for the first player in each diagram, and then used multiple analogous diagrams to show several representative choices for the second player. As such each diagram and the outcomes it depicts vary based on the norm employed by the second player. The general assumption of traditional economic theory is that humans are rational maximizers: players should make choices that maximize their payoff. If we assume that the UG is a one-shot game, the rational choice on the part of the first player is to offer a tiny fraction of the resource and the rational choice on the part of the second player is to accept that offer. The following image represents this scenario of the Ultimatum Game:
As with our sequence images for the SH, PD, and HD games, we represent the opportunity for each player to make a choice with a square. In the case of the UG, there are two kinds of squares, each representing one of the non-simultaneous choices made by our green (offering) and blue (choosing) players. Pie charts on the far left of the image show five representative offers that could be made by the green player, with five separate “choice” squares indicating these possibilities. The visual depiction of these choices is reinforced by a numerical representation of the split. The second player can only refuse or accept, and therefore has only two squares representing these two choices. Colored tubes connect each offer made by the green player to the choice made by the blue player based on the norm employed, which can be seen as a ‘thought bubble’ emanating from the blue player at the top of the image. If a refusal takes place, both players gain nothing, which is indicated by a pair of triangles representing the worst possible results. This worst possible result is reinforced by frowning faces on both players. For results that are accepted, the result experienced by each player is depicted using both distance of payoff and emotional response on each player’s face. Dashed vertical lines allow for the comparison of different payoffs. We call this version of our UG images “anything” because the second player is willing to accept any offer from the first player. To show the dynamics of the game, we had to assign this “anything” player a norm of ≥99/1, which technically does not accept all offers (it would reject a split of 99.5/0.5, but comes close enough to the “rational” ideal of accepting any and all offers).
Experimental work has shown that human players of the UG do not generally follow the “rational” strategy predicted by game theory. Instead, they maintain some higher sense of what is a fair offer, and are willing to sacrifice potential winnings to maintain this standard of fairness. To represent these higher thresholds for being willing to accept an offer, we created two additional norms, which we called “dignity” and “fair”. A player employing the “dignity” strategy may not feel entitled to a full half of the payoff at stake, but wants a large enough portion to maintain his dignity. What represents a ‘dignity preserving’ offer will of course vary from person to person and culture to culture, so we arbitrarily choose a norm of 65/35 to represent this “dignity” strategy. The sequence image for this norm is shown below:
The major difference between the “anything” and “dignity” norms lies in the range of offers which will be accepted. Whereas the “anything” player will reject only an offer of nothing, the “dignity” player rejects all offers that fall short of giving up 35% of the payoff. Accordingly, the image above shows that the offer of 90/10 is rejected, leading to the same result as the offer of nothing (100/0).
Essentially all norms contain an implied sense of what is fair and what is not fair, but in many cultures a strict egalitarian interpretation of “fair” is enforced via norms. In honor of this common norm, our last norm ”fair” rejects all offers that are worse than a 50/50 split:
Rejection now occurs for the first three offers, and the range of possible accepted offers shrinks.
As with other images we have produced, a series of these images allows the user to demonstrate the differences between different versions of the game. For the UG, the key comparison is between different norms, which produce different outcomes. The sequence below shows how these three norms can be lined up for comparison:
An even more effect way to demonstrate the difference between these three norms is to line them up sequentially as a series of slides in a presentation. Scrolling through the images causes the different results to ‘pop’ out at the audience.
We are looking to receive feedback on these images, in particular from those who teach game theory. Please feel free to contact us with your suggestions.