UPDATE: The images discussed below are now available for free use on the Evolutionary Games Infographic Project page.
To complete the set of Evolutionary Games Infographic images that Greg Riestenberg and I have been working on, we created a set of “conceptual” matrices for the Ultimatum Game (UG). These are meant to complement the conceptual images we have already created for the Prisoner’s Dilemma, the Hawk-Dove game, and the Stag Hunt game. As I have discussed previously, the UG is in several ways fundamentally different from the other three game constructs we have portrayed. This first fundamental difference is that the game is not symmetrical: the players make choices at different phases of the game, and the choices available to each player are different. The other fundamental difference is that the number of possible outcomes in the game is not finite; this arises from the non-discrete nature of the first player’s choices.
To acknowledge these differences while at the same time trying to make meaningful comparisons of these different classes of games, we chose to portray the UG in terms of offers and norms. The offer is the split of the resource suggested by the first player, and norm is the minimum portion of this split that will be accepted by the second player. To make the dynamics of the game tractable to explain, we selected five possible discrete offers and three possible discrete norms to display in our sequence graphics representing the UG. To complement this sequential depiction of the UG, we provide the following conceptual matrices:
In this matrix as with our sequence representations, the first player is depicted in green, with the word “offers” used to show that this is the player making the offer. The five offers are depicted on the columns with a pie chart showing the division of the payoff between the player who makes the offer (green) and the player who must decide whether or not to accept the offer based on his norm (blue). These pie depictions of the offers are reinforced with color-coded numerical splits. On the rows, three different norms for the second (blue) player are shown. Both the offers and the norms displayed in these matrices match the offers and the norms displayed in our sequence images. At the intersection of the offers and the norms, the matrix shows the resulting payoff for each player as a pie chart.
For offers below the norm, the second (blue) player refuses the offer, resulting in no payoff for either player. As with previous conceptual images, we show this worst-case outcome for both players using a light color intermediate to the blue and green colors that represent the individual players. For offers equal to or above the norm, the second (blue) player accepts the offer, resulting in the payoff scheme dictated by the original offer. These we show using the pie chart representing the offer, with the saturated color of the outlining box suggesting which player received the higher payoff. For offers favoring the green player, the outlining box is closer to green, whereas for offers favoring the blue player the outlining box is closer to blue. For the offer of 50/50, the color is intermediate.
We also show this same matrix with different depictions of the offers. Instead of showing the full blue/green pie for offers and norms, this image shows the offers and norms strictly based on what the blue player would receive:
The advantage of this depiction is that it focuses in on the key question of the UG, which is what the second player stands to receive through the offer and what that player conceives as a minimum acceptable offer via his personal norm.
A final way to represent this same matrix is to show the offers and norms from the perspective of each player:
Here the first player’s offer is shown based on what he stands to gain, depicted in the pie as green. The second player’s norm is shown based on what he would accept at minimum for himself, depicted in the pie as blue. All three versions of this basic matrix allow for the comparison of offer to norm and show the resulting payoff.
In line with our sequence images, we also show this same matrix using the facial expressions representing the relative and absolute outcomes for each player:
As with the ‘pie chart’ images, the outlining box provides a secondary clue as to which player received the greatest payoff.
The final image in our series provides this same matrix using only the numerical results:
Again, color is used to represent the relative and absolute outcomes, with the low-saturation intermediate color used to outline the “0/0″ result of a refusal by the second player, and various saturated shades ranging from blue to green depicting the results for offers that were accepted.
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.