The Diablo 3 Economy: An Agent Based Approach

Abstract

Designers of MMOs such as Diablo 3 face economic problems much like policy makers in the real world, e.g. inflation and distributional issues. Solving economic problems through regular updates (patches) became as important to those games as traditional gameplay issues. In this paper we provide an agent framework inspired by the economic features of Diablo 3 and analyze the effect of monetary policy in the game. Our model reproduces a number of features known from the Diablo 3 economy such as a heterogeneous price development, driven almost exclusively by goods of high quality, a highly unequal wealth distribution and strongly decreasing economic mobility. The basic framework presented in this paper is meant as a stepping stone to further research, where our evidence is used to deepen our understanding of the real-world counterparts of such problems. The advantage of our model is that it combines simplicity that is inherent to model economies with a similarly simple observable counterpart (namely the game environment where real agents interact). By matching the dynamics of the game economy we can thus easily verify that our behavioral assumptions are good approximations to reality.

Appendix: The Auction

1.1 Implementation

Although our model players are virtual agents themselves, they make use of bidding agents for computational reasons outlined below.

The bidding takes place in rounds, giving the players the chance to sequentially place their bids. In detail, a bidding round looks as follows. The sequence of players in a round only matters if players have identical willingness to pay and produces an outcome equivalent to truly simultaneous auctions otherwise. Since gold endowment is continuous, the probability of two players having precisely the same willingness to pay is zero.¹³

At the start of each bidding round $\tau$ of the auction, we check whether there is any player who cannot afford any item increasing his power, or for whom buying jewels only is the optimum choice given current prices. Since bids do not decline during an auction, any player meeting one of those criteria at any time in the auction, also meets those criteria for the remainder of the auction. Therefore, all those players leave the current auction permanently.

Because each player can only use one item, players who are currently highest bidders are not considered in the current round of bidding.

A player $n$ who is willing to bid first identifies the item that yields the highest power level considering the maximum jewel level that he or she can purchase from the remaining budget. Labeling the current bid for item $m$ in bidding round $\tau$ with $b_{\tau ,m}$, and $j^*(g_{t,n}-b_{\tau ,m})$ being the maximum jewel level available for his remaining $g_{t,n}-b_{\tau ,m}$ gold, the objective of the player $n$ thus is:

$$\begin{aligned} \psi (i^*_{t,\tau ,n},j^*_{t,\tau ,n}) = \max \limits _m{\psi (i_m,j^*(g_{t,n}-b_{\tau ,m}))}. \end{aligned}$$

(6)

The player can then compute a maximum bid $\bar{b}_{t,\tau ,n}$ that does not change his original considerations, i.e. the bid for $i^*$ where he or she can still afford $j^*$:

$$\begin{aligned} \bar{b}_{t,\tau ,n}=g_{t,n}-\tilde{P}(j^*_{t,\tau ,n},j_{t,n}). \end{aligned}$$

(7)

This price is communicated to the auction house as this players maximum bid.¹⁴ If the auction house receives bidding instructions for the same item by multiple players, it sets the new price to the second highest maximum bid plus one, remembers the highest bidding player and his maximum bid. All other players are informed that they have been outbid. While not affecting the outcome, the coordination by the auction house substantially reduces computational requirements since prices are not increased in small steps.

Since we find that very often the richest players compete for the best items, we start the auction in a limited setup that only includes the two richest players. When these two have reached a point where nobody is willing to place a further bid, the next player (ordered by available gold) enters the auction. Again, all players—including those who stopped bidding before—can bid until no one is willing to place another bid. This procedure is repeated until all players have entered the auction and no player is willing to place a higher bid. While yielding the same result as an auction where all players participate from the very beginning, this procedure reduces the number of bids that is necessary to reveal the second highest willingness to pay (i.e., the price) dramatically, thereby further decreasing runtime of the algorithm.

1.2 Dynamic auctions for heterogeneous goods—equalizing primal and dual solutions

Ausubel (2006) introduced this type of auction for heterogeneous and discrete goods. In short, agents report, for a given price vector, the quantities they would wish to purchase. Prices are then adjusted until final demand and supply equalize. The problem can also be expressed in a dual form. In that case, a bidder is credited units if the demand of opponents declines with rising prices, and he or she is debited units if the demand of opponents increases. Ausubel gives an example, in which the sum of credits and debits for every good equal final demand (i.e., the primal and dual solution) are the same. However, a small adjustment to his original approach has to be introduced to ensure that this is always the case in equilibrium: An initial credit or debit of every good $m$ needs to be assigned to player $n$, that is equal to the sum of initial demands of opponents (the total amount that can be credited to player $n$ during the auction) minus the final quantity of that good.

As described above, both the quantity and substitutability of goods is heavily restricted in our setup. These restrictions imply, that at starting prices ($b_{1,m}=i_m$ for every item $m$), players would only announce that they are interested in the item of the best quality (if they do not already possess an item of higher quality). To make sure, that the credit and debit assignments from the dual problem match the final outcome, an initial round and initial debt for every player can be added. First, every item in the auction gets an initial price $b_{0,m}=0$. Thus, every player would be interested in every item (as these items could afterward be sold for a profit to the in-game vendor). This creates initial excess demand of $N-1$ for every good. The dual procedure of the bidding process would assign in equilibrium a credit of $N-1$ units of every item to every of the $N$ players, plus an additional unit of the item the player actually purchases. Second, every player gets an initial debt, equal to $N-2$ unit of every item ($N-1$ units demanded by opponents and $1$ tradable unit). With these two initial conditions, our auction is not only equivalent to the one of Ausubel (2006), but also produces matching primal and dual solutions.