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I am trying to compute the (pure strategy) Nash equilibria of some discrete auctions.

More precisely, let us define the strategy of each player as a function mapping from every valuation that they might have to their bid (i.e. the 'bidding function'). Let us suppose that each player's valuation is drawn from some finite set and that their bid must belong to this same (finite) set. I am interested in finding the set of bidding functions, one for each player, such that each player's bidding function is optimal given the bidding functions of all the other players.

If it makes things easier, we can assume that valuations are symmetric (i.e. each player's valuation is generated by the same probability distribution) and that there are only two players. Ideally, however, we would proceed without these simplifications. I am interested in computing the equilibrium for the first price sealed bid and all pay auctions (in the latter, you pay your bid even if you lose; in the former, you don't.)

I have considered writing down the discrete auction game in normal form and finding the equilibria using software like Gambit. However, this would seem tricky since the strategy space is so large. For example, if a player chooses bids from $ \{1,...,10\} $ and draws values from $ \{1,...,10\} $, then already they have $10^{10}$ pure strategies.

Does anyone have any ideas about how to proceed here?

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    $\begingroup$ Thanks for these clarifications, now edited. $\endgroup$ – afreelunch Feb 17 at 21:51
  • $\begingroup$ Do you want to find Nash equilibria, or Bayes-Nash equilibria? $\endgroup$ – Ubiquitous Feb 17 at 22:25
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    $\begingroup$ Since you want to solve the game computationally, you'd have to control the dimension of your problem no matter what program you use. If $10^{10}$ pure strategies is too complex for Gambit, it's likely to be too complex for other software as well. At any rate, suppose you can limit the dimension of your problem to a feasible level. Then given a distribution of the player's types, you can easily compute the payoff matrix for each player in Matlab/R/Excel and then export the resulting matrices to Gambit to solve for any NEs. $\endgroup$ – Herr K. Feb 18 at 0:31
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    $\begingroup$ Thanks for these ideas. To be honest, I don't know if this payoff matrix is too large for Gambit. I guess I'll give it a go. Of course, one can significantly reduce the strategy space here by ruling out strategies that call you to bid above your valuation. $\endgroup$ – afreelunch Feb 18 at 16:01
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    $\begingroup$ @denesp: You're right. My method does not reduce the computational dimensions. Rather, it's a way to help Gambit solve for BNEs by converting the game into a Bayesian normal form (based on my experience with GTE, I suspect that Gambit does not do such a conversion). As I mentioned in my initial comment, the prerequisite is that the OP "can limit the dimension of your problem to a feasible level". $\endgroup$ – Herr K. Feb 18 at 20:17
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If $10^{10}$ pure strategies is too large for Gambit, it'll likely be too large for any other software as well. In the comments, I suggested that you could first compute the expected payoffs for each player in a separate program such as Matlab/R/Excel, and then export the resulting matrix to Gambit where the BNEs can be calculated. This way, you may be able to save some resources for Gambit on converting the game into its Bayesian normal form.

If you're okay with significantly reducing the dimension of your problem, then you can rely solely on Gambit (or, Game Theory Explorer (GTE), which is now a part of the Gambit project). In the following example done in GTE, I assumed that bidder $i$'s valuation and strategy spaces are binary: $V_i=S_i=\{0,1\}$ for $i=1,2$. The values are assumed to be uniformly distributed.

First, you enter the game in its extensive form. In the figure below, the four branches at the top correspond to Nature's choices that determine the valuations. From left to right, the branches represent the valuation profiles $(v_1,v_2)=(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$ respectively, each occurring with probability $1/4$.

enter image description here

After you finish entering the extensive form game, press the "Matrix Layout" tab to see the (automatically calculated) Bayesian normal form. There's a gear button at the top that will compute the BNE of the game.

enter image description here

The result looks like the following. In this case, the result was returned in seconds (in part thanks to the relatively small dimension of the problem). The unique BNE, in which each player bids zero regardless of their value, was found.

enter image description here

You can of course try to increase the size of the value/strategy space. This method would also allow you to specify asymmetric distributions of bidder values. However, GTE only solves two-player games. But Gambit seems to be able to solve games with more than two players.

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