📈 economics

The Moroccan Public Procurement Game

This paper analyzes the Moroccan public procurement market as a strategic game with discontinuous and non-quasiconcave payoffs, demonstrating the absence of pure-strategy Nash equilibria while establishing the existence of mixed-strategy equilibria for both two-player and general N-player scenarios.

Original authors: Nizar Riane

Published 2026-05-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nizar Riane

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Game of Guessing the Average

Imagine a group of people bidding on a contract to build a bridge. In the old days, the rule was simple: "Whoever offers the lowest price wins." But in 2023, Morocco changed the rules. Now, the winner isn't just the cheapest bidder.

Instead, the government sets a "Reference Price" (let's call it the Target). This Target is calculated by taking the average of everyone's bids and adding it to the government's own estimated cost.

The Goal: You want your bid to be the one closest to the Target, but not higher than it.
The Twist: If you bid too high, you lose. If you bid too low, you lose. You have to guess what everyone else will bid, calculate the average, and then aim your bid just below that average.

This turns the bidding process into a "Beauty Contest" (a famous economic concept). You aren't trying to be the most efficient; you are trying to guess what the group thinks the average will be, so you can position yourself just below it.

The Problem: No "Perfect" Move Exists

The author, Nizar Riane, uses Game Theory (the math of strategy) to analyze this new rule.

The Pure Strategy Problem:
Imagine you are playing this game with friends. You try to find one specific number to bid that will always win, no matter what your friends do.

If everyone bids the same amount, you can slightly lower your bid to win.
If everyone bids differently, you can calculate the perfect spot to undercut the average.
The Catch: As soon as you pick a "perfect" number, your friends can change their numbers to make your "perfect" spot a losing one.

The paper proves that in this game, there is no single "best" number to bid. If you try to stick to one specific number, you will eventually lose. In math terms, there is no "Pure Strategy" Nash Equilibrium (a stable state where no one wants to change their move).

The Solution: The Art of Randomness

Since you can't pick one winning number, the paper suggests the only way to play is to randomize. This is called a Mixed Strategy.

Instead of saying, "I will bid $100," you say, "I will randomly pick a bid between $90 and $110, following a specific pattern."

The paper finds two different ways to do this randomization for a game with two players:

The Uniform Mix: Imagine spreading your bids out like butter on toast over two specific intervals. You don't pick one spot; you spread your chances out evenly across two ranges.
The Functional Mix: Imagine a curve where you are more likely to bid certain numbers and less likely to bid others, following a specific mathematical formula (like a bell curve, but shaped differently).

Both methods result in a "fair" game where, on average, you win half the time (or a specific fraction of the time), and neither player can improve their odds by changing their random pattern.

What Happens When the Rules Get Uneven?

The paper also looks at what happens if the game isn't fair. What if Player 1's bid counts for 60% of the average, but Player 2's bid only counts for 40%?

The Metaphor: Imagine a tug-of-war where one team has a much longer rope. That team has more influence over the "Target."
The Result: The paper calculates exactly how the "winning probability" changes based on this imbalance. It shows that as one player gets more power to influence the average, the game shifts, and the winning odds change in predictable, mathematical ways.

The Big Group Problem (N Players)

Finally, the paper asks: "What happens if there are 10, 20, or 100 bidders?"

The Chaos: With more people, the math gets incredibly messy. The "Target" depends on the average of everyone, and the "winning zone" becomes a complex, jagged shape that is hard to describe.
The Mathematical Proof: The author couldn't write down a simple formula for 100 players like they did for 2. However, they used advanced math (specifically extending a famous theorem by Dasgupta and Maskin) to prove that a solution still exists.
The Takeaway: Even though the game is chaotic and the rules are "jagged" (discontinuous), there is still a stable, fair way for everyone to play using random strategies. The paper proves that a "Mixed Strategy Equilibrium" exists, even if we can't write down the exact formula for it in the general case.

Summary

The Game: A Moroccan bidding rule where you must guess the group average and bid just below it.
The Discovery: You cannot win by picking one specific number. You must randomize your bids.
The Math: The paper proves that while no "perfect" single move exists, there are specific patterns of random moves that create a stable, fair game for 2 players, and proves that such a stable game exists even for large groups, even if we can't easily calculate the exact pattern for large groups.

The paper is essentially a mathematical detective story showing that even in a chaotic, rule-breaking game, order and fairness can be found through the power of probability.

Technical Summary: The Moroccan Public Procurement Game

Problem Definition
This paper analyzes the Moroccan public procurement market following a March 2023 reform. Under the new rules, contracts are awarded not to the lowest bidder, but to the bid closest to a reference price $P$ , defined as $P = E + \bar{x}/2$ , where $E$ is the estimated cost and $\bar{x}$ is the average of all submitted bids. Bids must fall within an admissible interval $[A, B]$ . If no bid is below $P$ , the winner is the closest bid from above; ties are broken by lot.

The author models this mechanism as a non-cooperative $N$ -player game. The central challenge identified is that the payoff functions are discontinuous and non-quasiconcave. The outcome depends discontinuously on the relative positions of bids with respect to the aggregate average, violating the semi-continuity and convexity assumptions required for classical equilibrium existence theorems (e.g., Nash, Glicksberg). The game is characterized as an aggregative game where payoffs depend on the full strategy profile, not just the individual strategy and an aggregator.

Methodology
The paper employs game-theoretic tools specifically designed for discontinuous games to establish the existence and explicit form of equilibria. The methodology proceeds in three stages:

Pure Strategy Analysis: The author first demonstrates that no Nash equilibrium exists in pure strategies. This is shown by proving that for any profile, a player can always deviate to a specific bid (calculated based on opponents' bids and $E$ ) to secure a win, exploiting the sensitivity of the mean to individual bids.
Explicit Construction (Two-Player Case): For the symmetric two-player case, the paper derives two distinct mixed-strategy Nash equilibria using constructive methods:
- Uniform-Based Strategy: Inspired by Sion and Wolfe (1958), this approach constructs a value of $1/2$ using a mixture of uniform distributions over specific sub-intervals of $[A, B]$ .
- Functional Equation Solution: Following Dresher (1961), the author sets up a system of functional equations based on the optimality conditions. This yields a second equilibrium characterized by a density function proportional to $1/(x-E)$ over a specific support, distinct from the uniform-based solution.
General $N$ -Player Case: For $N \geq 3$ $N \geq 3$ , explicit construction becomes impracticable due to the complexity of the discontinuity sets. The author instead applies an extended version of the Dasgupta and Maskin (1986) framework. This involves:
- Defining discontinuity sets (ties, fixed points, and transition points) and showing they have zero Lebesgue measure.
- Verifying that the game satisfies weak lower semicontinuity and that the sum of payoffs is upper semicontinuous.
- Establishing payoff security in the mixed extension by verifying the disjoint payoff matching condition (Allison and Lepore, 2014), which overcomes the failure of uniform payoff security in this specific game structure.
- Proving that a sequence of finite-game equilibria converges to a mixed-strategy equilibrium in the continuous game.

Key Results

Non-Existence in Pure Strategies: The game admits no Nash equilibrium in pure strategies for any $N \geq 2$ .
Two-Player Equilibria: Two explicit symmetric mixed-strategy equilibria are derived:
1. A strategy involving a mixture of uniform distributions on intervals $[A, A_1]$ and $[A_1, A_2]$ , yielding a game value of $1/2$ .
2. A strategy with a density $f(x) \propto 1/(x-E)$ on $[A, A_2]$ , also yielding a value of $1/2$ .
Asymmetric Two-Player Case: The paper generalizes the model to a weighted reference price $P = (px + (1-p)y + E)/2$. It identifies different regimes based on the weight parameter $p$ (symmetric, intermediate, critical, and low- $p$ ). Explicit game values are derived, such as $1/3$ in the critical regime and $2/5$ in the intermediate regime, with explicit density functions provided for the general case.
Existence in General $N$ -Player Case: The paper establishes the existence of a symmetric mixed-strategy Nash equilibrium for any $N \geq 3$ . This is achieved by extending the Dasgupta-Maskin result to handle the specific discontinuity structure of the procurement game, proving that the mixed extension is better-reply secure.

Significance and Claims
The paper claims to provide a rigorous theoretical foundation for understanding the new Moroccan procurement mechanism, which transforms the market into a "Keynesian beauty contest" where bidders anticipate collective behavior rather than minimizing costs.

The primary contribution is the constructive derivation of explicit equilibrium strategies for the two-player case, a rarity in the literature on discontinuous games which often relies on non-constructive existence proofs. Furthermore, the paper extends the Dasgupta-Maskin (1986) existence theorem to a specific class of aggregative games with non-standard discontinuities (dependent on the full profile rather than just the aggregator), thereby establishing the existence of symmetric mixed equilibria for the general $N$ -player case. The work bridges the gap between abstract discontinuous game theory and a concrete, real-world economic policy reform.