Mover in Grid Games: Ambiguity & Optimal Paths

• Mover is the informed, goal-directed player in a grid game with asymmetric information, using ambiguous moves to conceal its true objective.
• The strategy involves front-loading ambiguous moves to delay the opponent’s inference and reduce resource consumption at the true goal.
• Equilibrium analysis shows that following shortest paths without exaggeration yields optimal outcomes against an adversarial Eater.

In the sense formalized in “The Eater and the Mover Game,” the Mover is the informed, goal-directed player in a two-player dynamic game with asymmetric information. Nature specifies two candidate goals $g_1,g_2\in \mathbb{Z}^2$, exactly one of which is the true goal $g_R$; the Mover knows which one is real, while the Eater does not. The Mover must physically move on the grid until it reaches the true goal and seeks to minimize resource consumption at that goal before arrival. The central result is that, under the paper’s game model and the Eater’s worst-case objective, the Mover can benefit from deception by ambiguity—hiding which goal is real for as long as possible—but not from deception by exaggeration—deliberately moving toward the fake goal to suggest false intent (Rostobaya et al., 2023).

1. Formal role of the Mover

The game’s information structure is asymmetric and explicit. The set of candidate goal locations

$G=\{g_1,g_2\}$

is common knowledge. At time $t$, the Mover observes its own position $p(t)$, the resource-consumption state $b(t)$, and private information

$I=\{1,2\},$

indicating which candidate is the true goal. Thus the Mover knows the true goal $g_R$, while the Eater observes motion and acts without knowing whether the game is $G_1$ or $G_2$, where $g_R$0 means that $g_R$1 is the true goal (Rostobaya et al., 2023).

The distinction among the publicly known candidate goals and the privately known true goal is fundamental. The notation is:

• $g_R$2: the two candidate goals,
• $g_R$3: the true goal,
• $g_R$4: the fake goal.

The Mover’s task is to retrieve the remaining bananas at the true goal when it arrives there. If $g_R$5 denotes the amount consumed from goal $g_R$6 up to time $g_R$7, then when $g_R$8 is the true goal the relevant loss is $g_R$9, where $G=\{g_1,g_2\}$0 is the arrival time. Accordingly, the outcome of game $G=\{g_1,g_2\}$1 under strategies $G=\{g_1,g_2\}$2 is

$G=\{g_1,g_2\}$3

and the Mover’s objective is simply

$G=\{g_1,g_2\}$4

The Mover therefore does not infer the true goal; it optimizes motion given private knowledge of that goal.

2. Sequential game model and signaling through motion

The game state has two components. The Mover position is

$G=\{g_1,g_2\}$5

The banana-consumption state is

$G=\{g_1,g_2\}$6

where $G=\{g_1,g_2\}$7 records consumed, not remaining, bananas (Rostobaya et al., 2023).

The action spaces are discrete. The Mover acts in

$G=\{g_1,g_2\}$8

while the Eater acts in

$G=\{g_1,g_2\}$9

corresponding to eating one banana from $t$0, one from $t$1, or half from each. Banana consumption evolves as

$t$2

where $t$3, and the terminal time is

$t$4

The Eater acts last at time $t$5, and then the Mover receives the bananas remaining at $t$6.

The timing is sequential: at each time $t$7, the Mover chooses a move, the Eater observes the most recent move, the Eater chooses where to eat, and the banana state updates. This makes the Mover’s trajectory the game’s signaling channel. Formally, for game $t$8 the Mover strategy is

$t$9

with overall strategy

$p(t)$0

while the Eater strategy is

$p(t)$1

The Eater thus knows the current position, the most recent Mover action, and the banana-consumption vector, but not which candidate is the true goal.

3. Geometry of Mover strategies

The paper classifies Mover actions geometrically using Manhattan distance. Let

$p(t)$2

be the $p(t)$3-norm, and define

$p(t)$4

The distance change induced by the Mover’s action is

$p(t)$5

A move is ambiguous if

$p(t)$6

so it does not reveal a preference for either candidate goal. A move is explicit if

$p(t)$7

so it reveals directional preference by moving toward one goal and away from the other. Since $p(t)$8, every move is one of these two types (Rostobaya et al., 2023).

For any current position $p(t)$9, the paper defines:

• $b(t)$0: the minimum number of ambiguous moves needed to reach a goal,
• $b(t)$1: the minimum number of explicit moves needed to reach $b(t)$2.

A shortest path to $b(t)$3 has total length

$b(t)$4

and all shortest paths have the same numbers of ambiguous and explicit moves, even if the exact path differs. This yields a clean decomposition of shortest-path geometry into delayable ambiguity and unavoidable commitment.

Within this geometry, the paper contrasts two forms of deceptive motion. Deception by ambiguity means hiding the true goal by choosing moves that preserve uncertainty for as long as possible; in the game, this means executing ambiguous moves first. Deception by exaggeration means making the fake goal look real, for example by moving explicitly toward $b(t)$5 early. The theoretical analysis shows that the Mover should not sacrifice path efficiency for deception: it should remain on shortest paths and exploit only the ambiguity already present in shortest-path structure (Rostobaya et al., 2023).

4. Equilibrium and the Mover’s optimal structure

The equilibrium concept is defined directly through the players’ optimization problems rather than through a Bayesian estimator. A strategy pair $b(t)$6 is an equilibrium if, for all admissible deviations,

$b(t)$7

The first two inequalities make the Mover optimal in each possible true-goal game; the third makes the Eater optimal given the Mover strategy pair (Rostobaya et al., 2023).

The proposed equilibrium Mover strategy $b(t)$8 is structurally simple:

• move on a shortest path to $b(t)$9,
• make all $I=\{1,2\},$0 ambiguous moves first,
• then make all $I=\{1,2\},$1 explicit moves.

Theorem 1 states that $I=\{1,2\},$2 and $I=\{1,2\},$3 form an equilibrium. The Mover-side proof is organized by several lemmas. One lemma shows that the Mover has no incentive to deviate from a shortest path by making ambiguous moves that go away from both goals. A second, labeled No Exaggeration, states:

The Mover has no incentive to make any explicit moves towards $I=\{1,2\},$4 if the Eater uses $I=\{1,2\},$5.

The proof argument is that an explicit move toward the fake goal can at best reduce later ambiguous-move consumption by at most $I=\{1,2\},$6, but it necessarily increases the number of explicit moves still required to reach the true goal, and each such remaining explicit move causes $I=\{1,2\},$7 unit of consumption from the true goal. Exaggeration can therefore at best cancel its own penalty and never improve total payoff.

A corollary is that the Mover has no incentive to use a non-shortest path if the Eater uses $I=\{1,2\},$8. A further lemma shows that, among shortest paths, the Mover has no incentive to deviate from making all ambiguous moves first. This sharpens the equilibrium claim: optimal deceptive motion is not merely shortest-path motion, but shortest-path motion with ambiguity maximally front-loaded. A common misconception is that deceptive motion should maximize false intent signaling; here, the equilibrium result instead shows that hiding intent is effective, while acting out false intent is not (Rostobaya et al., 2023).

5. Mathematical characterization of the Mover’s advantage

A central set of auxiliary quantities is

$I=\{1,2\},$9

The paper interprets $g_R$0 as the value that would arise under complete information if the Eater knew the true goal and always consumed there. It also defines the differences

$g_R$1

where

$g_R$2

These difference terms govern the Eater’s conservative behavior during ambiguous motion and thereby determine the Mover’s incentives (Rostobaya et al., 2023).

During ambiguous moves, the paper states

$g_R$3

and under $g_R$4,

$g_R$5

Thus, under ambiguity and worst-case Eater play, the difference term moves one step toward zero. During explicit moves toward $g_R$6, the paper gives

$g_R$7

Once the Mover commits directionally, the Eater responds by consuming from the revealed goal.

The equilibrium outcome is given by Theorem $g_R$8. Suppressing time arguments, the paper states

$g_R$9

Because of PDF corruption, the exact symbols are partially damaged in the source text, but the theorem’s structure is clear from the proof and surrounding notation. The Eater’s equilibrium value is then

$G_1$0

The paper explicitly interprets the subtracted terms,

$G_1$1

as the reduction in consumption relative to the complete-information scenario. This is the precise mathematical expression of the Mover’s informational advantage: ambiguity reduces true-goal consumption without requiring extra path length.

6. Numerical illustration and practical interpretation

The numerical section visualizes the equilibrium values $G_1$2, $G_1$3, and $G_1$4, and highlights regions where the Eater’s worst-case value switches between the two possible true-goal games. The most direct illustration of the Mover’s behavior is Figure 1, which compares three trajectories against the equilibrium Eater strategy $G_1$5 (Rostobaya et al., 2023).

The three trajectories are:

1. Path (ii): equilibrium shortest path with ambiguous moves first The Eater consumes 4 bananas from the true goal.
2. Path (i): shortest path, but explicit moves before ambiguous ones The Eater consumes 5 bananas.
3. Path (iii): exaggeration strategy moving toward the fake goal first The Eater consumes 6 bananas.

This numerical example mirrors the theory exactly. Ambiguity helps the Mover, the ordering of shortest-path moves matters, and exaggeration hurts. More broadly, the formulation differs from deceptive-motion models that measure deception through the quality of an external observer’s inference. Here deception is measured through the outcome of the game itself: the relevant question is not whether an observer guesses incorrectly, but whether the opponent’s actions consume fewer bananas at the true goal.

The paper’s practical lesson is correspondingly narrow but clear. Against an active, adversarial observer that optimizes for the worst case, the Mover should exploit naturally available ambiguity in geometry, preserve shortest-path efficiency, delay commitment until ambiguity is exhausted, and avoid explicit motion toward a decoy goal. A plausible implication is that, in adversarial signaling settings of this type, robust deceptive motion is not primarily about inducing false belief; it is about constraining the opponent’s action policy under asymmetric information.