Tóth's food serving game

Recently I've dived into game theory by attending a course. I've always been haunted by a dilemma while serving food to my guests at my apartment and now I'd like to introduce this problem in the formal terms of game theory - just for fun :D

Let's have a grill party G(N, f), where N is the set of hungry participants (|N| is the number of them), and f is the amount of food available that is divided among p₁, p₂, ... p_|N| plates. Let h ∈ N be the host of the party, g₁, g₂, ..., g_|N|-1 ∈ N the guests. h serves the food for all the participants including h, as well as divides the f amount among the plates while, being a kind host, lets the guests to choose which plate they want to take. In the end h takes the last one left. Actions of the players' are as follows (a₁, a₂, ..., a_n ∈ A_x denotes the set of actions player x is able to take):

• A_h: h is able to decide the amount of food he puts on each plate. Therefore, a_i ∈ A_h is the distribution of food among the plates that is (p₁, p₂, ..., p_|N|) – each action is a tuple of plates. Assuming that h has no priority of giving more food to g_i than to g_j (i ≠ j), h is able to decide the proportion of food he puts on one plate and the average of the food amount on all other plates: (p₁, average_i(p_i)), a 2-element tuple where i ≠ 1. It is by definition that p₁ + (|N| - 1) · average_i(p_i) = f, i ≠ 1. Here A_h can be simplified to 3 actions if we neglect the exact proportion of p₁ and average_i(p_i):
  • a₁: p₁ > average_i(p_i),
  • a₂: p₁ = average_i(p_i),
  • a₃: p₁ < average_i(p_i).
• A_gi: g_i is able to choose which p_k plate to take - a_gi = k: g_i takes p_k. The last plate left goes to h automatically.

The payoff u(player_i, A) for each player_i ∈ N is the amount of food taken considering the set of actions A taken by all players – if g_i takes the k_th plate (a_gi = k), then u(g_i, A) = p_k. Now, if h distributes food in a unequal way, h may speculate that the last plate it gets contains more food in ratio to the average. On the contrary, if all the "good" plates are gone, h suffers from being selfish and receives less than f / |N|. The Nash equilibrium of the game intuitively is p₁ = average_i(p_i) and a_gi is taking an arbitrary plate, as in this case p₁ = p₂ = ... = p_|N|. To put it simple, let's see a |N| = 2 case with one guest and the host playing, let f = 4, A_g = {1, 2}, A_h = {(1,3), (2,2), (3,1)} by neglecting the continuous nature of A_h (Nash equilibria are indicated by red bordering):

Host	Guest
	Ah\Ag	1	2
	(1,3)	3,1	1,3
	(2,2)	2,2	2,2
	(3,1)	1,3	3,1

Let's see an example from the matrix: if h puts 1 unit food on p₁, and 3 units on p₂ (a_h = (1,3)), and if g then chooses to take p₂ (a_g = 2), then u(h, {a_h, a_g}) = 1, because h gets p₁ (the one plate left) containing 1 unit of food, and u(g, {a_h, a_g}) = 3, because g gets p₂ (what was chosen by it) containing 3 units of food. In this case we're in the 1_st row and 2_nd column of the game matrix: (1,3).

The general case matrix with arbitrary number of guests and continuous A_h would have a |N| dimensional representation where the first dimension is infinite – that is determined by the infinite number of choices of h. Each cell of such a matrix would contain:
(p_left, p_ag1, p_ag2, ..., p_ag|N-1|) where p_left ∉ {p_ag1, p_ag2, ..., p_ag|N-1|}, and p_agi is the plate g_i chooses.

The moral of the game is: you'd better distribute that food equally, because it always converges to the Nash equilibrium in the long term – alternatively you can just eat some before serving without anyone to notice :3